Universal Bounds for Codes and DesignsVladimir I. Levenshtein�Contents1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Codes and designs in compact metric spaces . . . . . . . . . . . . . . . . . . 52.1 Parameters of codes in compact metric spaces . . . . . . . . . . . . . . 52.2 A system Q of orthogonal polynomials for a compact metric space . . 122.3 A restricted � -design problem for systems of orthogonal polynomials . 153 Polynomial metric spaces and extremum problems for the system Q of or-thogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1 Inequalities for nonnegative de�nite functions . . . . . . . . . . . . . . 213.2 Polynomial metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 A �-packing and � -design problem for systems of orthogonal polynomials 434 Duality in bounding optimal sizes of codes and designs in polynomial graphs 514.1 Polynomial graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Basic inequalities for code parameters based on annihilating polynomials 604.3 Duality in bounding optimal sizes of codes and designs . . . . . . . . . 675 Applications of orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . 725.1 Properties of orthogonal polynomials . . . . . . . . . . . . . . . . . . . 725.2 Bounds on extreme roots of orthogonal polynomials . . . . . . . . . . 775.3 Properties of adjacent systems of orthogonal polynomials . . . . . . . 845.4 Main theorem and consequences . . . . . . . . . . . . . . . . . . . . . 885.5 Applications to polynomial metric spaces and polynomial graphs . . . 1006 Summary for the basic polynomial spaces . . . . . . . . . . . . . . . . . . . 1096.1 The unit Euclidean sphere and the projective spaces . . . . . . . . . . 1096.2 The Hamming space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.3 The Johnson space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129�The research was partially supported by the Russian Foundation for Basic Research under grant95-01-01103 and by the International Science Foundation under grant MEF300.

1. IntroductionThis chapter deals with packing and some related problems for a class of polynomialmetric spaces X = (X; d(x; y)) including �nite and compact in�nite ones, in partic-ular, Hamming space, Johnson space, the unit Euclidean sphere, and real, complexand quaternionic projective spaces. We are interested in bounds on parameters of�nite subsets (codes) C of X which are universal in the following sense. First thebounds are valid for any code C � X as distinguished from the existential boundsvalid only for some codes. Moreover they are valid for any polynomial metric spaceX , in particular, for the above mentioned metric spaces. The method considered forobtaining universal bounds is based on a description of all nonnegative de�nite func-tions F (x; y) on X which depend only on the distance d (x; y). Every such functiongives a universal upper bound on the size jCj of a code C � X with minimal distanced (C) between its distinct points. We want to choose such a nonnegative de�nitefunction which gives rise to the best bound.For a polynomial metric space X there exists a system Q = fQi (t) ; i = 0; 1; : : :gof orthogonal polynomials in a real t and a certain decreasing function � (d) = �Q (d)such that for a polynomial f (t), the function f (� (d (x; y))) is nonnegative de�niteon X if and only if f (t) can be expanded over the system Q with nonnegative coef-�cients. In the framework of the method being considered, this reduces the problemof obtaining the best upper bound on the size of a code with minimal distance d to acertain extremum problem (�-packing problem, � = � (d)) for the system Q of orthog-onal polynomials. The author [64] found the optimal solution (over polynomials ofrestricted degree) of the �-packing problem and the corresponding objective functionLQ(�) which is an increasing continuous function in �. This implies the followinguniversal bound for an arbitrary code C � X :jCj � LQ (�(d (C))) (1.1)and gives necessary and su�cient conditions for its attainability which are satis�edin many cases. Any code C with equality in (1.1) is distance-regular and all distancesbetween code points and intersection numbers are uniquely determined by d(C) andsome parameters of the whole space X . To a certain extent this chapter is devotedto the description and substantiation of (1.1) for a polynomial metric space X and tocalculations of (1.1) for di�erent spaces X of signi�cant interest in coding theory.Following Delsarte [29], [30] we will also characterize a code C � X by its dualdistance d0 (C) and/or the maximal strength � (C) = d0 (C) � 1 of a design formedby C. In order to obtain a universal lower bound on the size of a code with a givendual distance d0 we consider one more extremum problem (the � -design problem,� = d0 � 1) for the system Q. For any � = d0 � 1 using the optimal solution ofthe �-packing problem for some special � = (d0) ; we obtain an optimal solution ofthe � -design problem over polynomials of degree at most � and, as a result, anotheruniversal bound for a code C � X :jCj � LQ ( (d0 (C))) ; (1.2)2

where (d) = Q (d) is a certain increasing function in an integer d. In particular,(1.2) gives the Rao bound [92] for orthogonal arrays in the case of the Hammingspace, the Ray-Chaudhuri-Wilson bound [93] for classical block designs in the case ofthe Johnson space, and the Delsarte-Goethals-Seidel bound [35] for spherical designsin the case of the unit Euclidean sphere. We extend the concept of dual distanceand maximal strength of a design to the case of a weighted set and of an arbitrarycompact metric space (not necessarily polynomial) and prove the bound (1.2) for thegeneral case. If we denote the maximum size of a code C � X with d (C) � d byA (X; d) and the minimum size of a weighted set C � X with d0 (C) � d by B (X; d) ;we can rewrite (1.1) and (1.2) as followsA (X; d) � LQ (�Q(d)) and B (X; d) � LQ ( Q (d)) :For the case of �nite polynomial metric spaces X which are P - and Q-polynomialassociation schemes [30] (called polynomial graphs in this chapter) we can considerthe �-packing and � -design problems for both �nite systems Q and P of orthogonalpolynomials. If we denote by AQ (d) and BQ (d) respectively objective functions ofoptimal solutions of the discrete �-packing (� = �Q (d)) and the discrete � -design(� = d� 1) problems for the system Q, we haveA (X; d) � AQ (d) � LQ (�Q(d)) ; (1.3)B (X; d) � BQ (d) � LQ ( Q (d)) : (1.4)For the system P we can de�ne the analogous functions AP (d), BP (d), �P (d), P (d)and LP (�) such that AP (d) � LP (�P (d)) ; (1.5)BP (d) � LP ( P (d)) : (1.6)In order to use (1.5)-(1.6) for obtaining universal bounds, we prove the followingduality relationship AQ (d)BP (d) = AP (d)BQ (d) = jX j : (1.7)This allows us to obtain from (1.3)-(1.7) two other universal bounds for a code C ina P - and Q-polynomial association scheme X :jCj � jX jLP ( P (d (C))) (1.8)and jCj � jX jLP (�P (d0 (C))) : (1.9)The bound (1.8) coincides with the Hamming bound when d (C) is odd. The bound(1.9) seems to be new; it was published by the author in [71] for Hamming space.3

Thus for �nite polynomial metric spaces we have two pairs (1.1)-(1.2) and (1.8)-(1.9)of universal bounds for codes and designs. For a number of spaces combinatorialproofs of (1.2) and (1.8) are known; however it is not likely that similar proofs of(1.1) and (1.9) can be found.For polynomial metric spaces under consideration we also have one more universalbound jCj � LQ ( Q (2s (C) + 1� � (C))) (1.10)where s (C) is the number of di�erent distances between distinct points of C and� (C) = 1 if the diameter of C is equal to the diameter of X and � (C) = 0 otherwise.This bound coincides with the absolute bound of Delsarte [30] when � (C) = 0. For�nite polynomial metric spaces a certain dual analog of (1.10) is valid.The chapter is organized as follows. In Section 2 we de�ne the basic parameters ofcodes and, in particular, present a concept of a (weighted) design in a compact metricspace. For a compact metric space we construct a system Q of polynomials which areorthogonal with respect to a measure on [�1; 1] connected with the average measureof metric balls. This system allows us to study some properties of codes and obtaina universal bound for the size of designs in compact metric spaces using the optimalsolution of a known extremal problem for systems of orthogonal polynomials. InSection 3 we study inequalities for nonnegative de�nite functions, introduce a notion ofpolynomial spaces, and describe their fundamental properties. The �-packing and � -design problems for the systems Q corresponding to polynomial spaces are formulatedand a preliminary description of a solution of the restricted �-packing problem isgiven. Elements of the Delsarte theory of polynomial graphs (P and Q-polynomialassociation schemes) are included in Section 4. The approach connected with usingadjacent systems of orthogonal polynomials to represent annihilating polynomials forcodes and designs turned out to be very useful in obtaining basic inequalities for codeparameters. In this section a new result on the duality in bounding optimal sizes ofcodes and designs is presented and a new pair of universal bounds for an arbitrarypolynomial graph is given which can be attained for codes which are generalizations ofMDS-codes and Steiner systems. Section 5 is devoted to the application of orthogonalpolynomials to solve problems of coding theory and is one of the main sections in thischapter. Numerous properties of adjacent systems of orthogonal polynomials aredescribed which are necessary to prove the main theorem on the optimal solutionof the restricted �-packing problem. This solution seems to be new for the theoryof orthogonal polynomials as well. We use a non-traditional approach to estimateextreme roots of orthogonal polynomials which are needed for obtaining asymptoticresults. In Section 6 we apply the theory described and calculate the universal boundsfor metric spaces of essential interest in coding theory: the unit Euclidean sphere, theprojective spaces, Hamming space, and Johnson space.4

2. Codes and designs in compact metric spaces2.1. Parameters of codes in compact metric spacesWe consider a �nite or compact, in�nite metric space X with distance d(x; y). Any�nite subset (code or design) C;C � X; is characterized by its minimal distanced(C) = minx;y2C;x6=y d(x; y)when jCj � 2 and by its covering radius�(C) = maxx2X d(x;C);where d(x;C) denotes the minimal distance between x and points of C. Classicalproblems are to �nd for any d > 0A(X; d) = maxC�X;d(C)�d jCj (2.1)and K(X; d) = minC�X;�(C)�d jCj:We assume that X is endowed with a normalized measure �, � (X) = 1, such that forany � (� � 0) and any x 2 X the metric ballS�(x;X) = fy : y 2 X; d(x; y) � �g (2.2)is measurable and �(x; �) = �(S�(x;X)) > 0 if � > 0: (2.3)In the in�nite case we assume that �0�(x; �) = @�(x;�)@� exists and is continuous as afunction in two variables. In the case of �nite X we assume that � is the normalizedcounting measure, that is, � (A) = jAjjX j for any A � X:The space X is called distance invariant if �(x; �) does not depend on x 2 X for any� . In the general case the function� (�) = ZX �(x; �)d� (x) (2.4)characterizes the mean measure of a metric ball of radius �. Thus, for distanceinvariant spaces, � (�) = �(x; �) for any x 2 X .5

We also consider some other parameters of codes connected with their metricproperties. For any W;W � X; let �(W ) denote the set of values of d(x; y) whenx; y 2 W and let �0(W ) be �(W )nf0g: For any �nite set (code) C;C � X; theparameter s(C) = j�0(C)j (2.5)characterizes the number of (distinct) distances between distinct points of C. Inparticular, s(X) is de�ned in the case of a �nite metric space X . In the case ofcompact in�nite metric spaces X; we put s(X) = 1 since then the set �(X) isin�nite. Clearly, for any code C,d(C) = min�0(C): (2.6)Together with the diameter D(X) = max�0(X) of the whole space X , we de�ne thediameter of a code C by D(C) = max�0(C):We also consider the following auxiliary parameter of a code C:�(C) = � 1 if D(C) = D(X);0 otherwise. (2.7)We call a code C with �(C) = 1 diametrical. For any code C � X; any x 2 X andany �, the set S�(x;C) = fy : y 2 C; d(x; y) = �gis �nite. Let B�(x;C) = jS�(x;C)j: (2.8)We call a code C � X distance invariant if for any �; B�(x;C) does not depend onx 2 C (i.e., if C is a distance invariant metric space) and completely distance invariantif for any x 2 X and for any �; B�(x;C) depends only on d(x;C). For a code C � X;let �(C) = fd0 = 0; d1; :::; dsg where s = s(C):We call a code C � X distance-regularif for any i; j 2 f0; 1; :::; sg and for any x; y 2 C the numberspx;yi;j = jSdi(x;C)\Sdj (y; C)j (2.9)depend only on d(x; y):We denote the intersection numbers (2.9) for a distance-regularcode C by pki;j if d(x; y) = dk . This de�nition of a distance-regular code may be alsoapplied for a �nite metric space X if we put C = X: Note that for �nite spaces X;(2.4) can be rewritten as follows� (�) = 1jX j2 Xx2X Xdi��Bdi(x;X); (2.10)and hence � (�) is right continuous (i.e., � (�+ 0) =� (�)).6

Now we introduce one more fundamental parameter of a code C and the conceptof a � -design (� = 0; 1; : : :) : For some applications it is useful to consider a �nite set(code or design) C as a weighted set C = (C;m) where m is a certain positive-valuedfunction on C (for example, multiplicity or probability). It is convenient to normalizethe weights m(x) of elements x such thatXx2Cm (x) = jCj ; (2.11)where jCj as always is the number of distinct elements (the size) of a weighted setC = (C;m): Hereafter we consider a code C as a special case of a weighted set whenm (x) � 1 for all x 2 C: The concept of a (weighted) � -design depends on a continuousstrictly monotone real function (substitution) � (d) de�ned on the interval [0; D(X)](for example, �(d) = d; d2 or ed): A weighted set C = (C;m) will be referred to asa weighted � -design (in X with respect to the substitution � (d)) if for any polynomialf (t) in a real t of degree at most � ,ZX ZX f(�(d(x; y)))d� (x) d�(y) = 1jCj2 Xx;y2C f(�(d(x; y)))m (x)m(y): (2.12)A � -design is a special case of a weighted � -design when m (x) � 1 for all x 2 C:Such � -designs are also called simple. The maximum integer � (� � s(X)) such thata (weighted) set C is a (weighted) � -design is called the strength of C and denotedby �(C): The value d0(C) = �(C) + 1 (2.13)is referred to as the dual distance of C: These de�nitions of a � -design and of thedual distance are natural extensions to compact metric spaces of the correspondingde�nitions of Delsarte for association schemes [30] and Delsarte, Goethals and Seidelfor the Euclidean sphere [35]. They coincide with the classical de�nitions for corre-sponding substitutions (linear for Hamming and Johnson spaces and quadratic for theEuclidean sphere). Some other approaches to extending designs can be found in [46],[83], [36], [84], [25]. The equality (2.12) shows that in a de�nite sense a (weighted)� -design C is a good approximation to the whole space X and the parameter (2.13)characterizes the degree of such an approximation. This explains why the problem of�nding B(X; d) = minC�X;d0(C)�d jCj (2.14)is of signi�cant interest in factorial experiments, cryptography, complexity theory andapproximation theory ([19], [92], [45], [112], [2], [48], [98], [28]). It should be notedthat the minimum in (2.14) is taken over all weighted (not only simple) (d�1)-designsC = (C;m).For values A(X; d) and K(X; d) there exist so-called sphere packing bounds basedon the facts that the open metric balls of radius d(C)=2 circumscribed around all7

points of a code C do not intersect and henceXx2C �(x; d(C)2 � ") � �(X) = 1 for any " > 0; (2.15)and the closed metric balls of radius �(C) circumscribed around all points of C coverX and hence Xx2C �(x; �(C)) � �(X) = 1: (2.16)In particular, for a distance invariant metric space X; (2.15)-(2.16) imply thatA(X; d) � (�(d2 � 0))�1; (2.17)K(X; d) � (�(d))�1:We shall consider a method of obtaining universal bounds based on nonnegativede�nite functions on X which allow us to improve (2.15) for all codes C with su�-ciently large d(C): On the other hand, the inequality (2.16) can be rather precise forcodes with both small and large covering radius. The author does not know resultsimproving (2.16) by this method (the interested reader can �nd bounds on K(X; d)in chapter (Brualdi, Litsyn, Pless)). However, this method gives rise to good lowerbounds on B(X; d) and shows that in a certain sense the design problem (similar tothe covering problem) is dual to the packing problem.Example 2.1. The Hamming space X = Hnv (n; v = 2; 3; :::) consists of vectors x =(x1; :::; xn) where xi 2 f0; 1; :::; v � 1g with the distance d(x;y) being equal to thenumber of coordinates in which x and y di�er. In this case �(X) = f0; 1; :::; ng andhence s(X) = n; D(X) = n: The Hamming space is distance invariant and�(d) = v�n X0�i�d�ni�(v � 1)i:The value A(Hnv ; d) is the maximum size of a code C � Hnv with d(C) � d; i.e., acode C capable of correcting bd=2c or fewer additive errors. For odd d = 2h+ 1 wehave �(d2 � 0) = �(h) and (2.17) gives the well known Hamming bound. It is known[29] that a � -design C with respect to a linear substitution �(d) is nothing but anorthogonal array of strength � and of index � = jCjv�� with n factors and v levels.This object is de�ned as a matrix C with rows from Hnv such that every vector ofH�v occurs in any � columns of C exactly � times. The value B(Hnv ; � + 1) is theminimum size of (weighted) � -designs in the Hamming space Hnv :Example 2.2. The Johnson space Jnw (n = 2; 3; ::: ;w = 1; :::; bn2 c) is the set of allw-subsets of the n-set f1; :::; ng ; where the distance between two elements x; y 2 Jnw8

is de�ned by w � jx \ yj: In this case �(X) = f0; 1; :::; wg and hence s(X) = w;D(X) = w: The Johnson space is distance invariant and�(d) = �nw��1 X0�i�d�wi��n� wi �:A set C is a � -design in Jnw with respect to a linear substitution �(d) (see [30]) ifand only if it is a classical block design S�(�; w; n) and � = jCj�w� �=�n�� (S�(�; w; n) isde�ned as a set S of w-subsets of an n-set such that each � -subset of the n-set belongsexactly to � w-subsets of S). The Johnson space Jnw can also be considered as thesubset of the binary Hamming space Hn2 consisting of vectors which have exactly wnon-zero coordinates, with distance being equal to half of the Hamming distance.Example 2.3. The unit Euclidean sphere in RnSn�1 = (x = (x1; :::; xn) 2 Rn; nXi=1 x2i = 1)with the Euclidean distance d(x;y) = pPni=1(xi � yi)2 is a compact metric space.In this case �(X) = [0; 2] and hence D(Sn�1) = 2; s(X) =1: We can also measurethe distance between x;y 2 Sn�1 by the angle '(x;y) wherecos'(x;y) = nXi=1 xiyi = 1� 12d2(x;y): (2.18)The following facts are well-known [26], [95], [42]. The isometry group G of Sn�1consists of all orthogonal matrices of order n and acts transitively on Sn�1. Thereexists a unique normalized measure � on Sn�1 which is invariant (i.e., �(gA) = �(A)for any measurable A � Sn�1 and any g 2 G). This measure � coincides with thenormalized Lebesgue measure on Sn�1 (the normalized surface area). The metricspace Sn�1 is distance invariant and for any d, 0 � d � 2,�(d) = �n�1(')�n�1 if cos' = 1� d2=2; (2.19)where �n�1(') is the surface area of a spherical cap on Sn�1 of radial angle ' (i.e.,of the set �y : y 2 Sn�1; '(x;y) � ' with x 2 Sn�1) and �n�1 = 2�n�1(�2 ) is thesurface area of Sn�1: It is also known (see, for example, [40]) that�n�1(')�n�1 = �(n2 )�(n�12 )�( 12 ) 1Zcos' (1� z2)n�32 dz (2.20)and �n�1 = 2� n2 =�(n2 ), where �(x) is the gamma function (�(x) = 1R0 vx�1e�vdv): Inthe case of X = Sn�1 the concept of a (weighted) � -design C = (C;m) is connected9

[35] (see also (3.58)) with the approximation formula for the evaluation of multi-dimensional integrals over Sn�1 of the following sortZSn�1 u(x)d�(x) � 1Px2Cm(x) Xx2C u(x)m(x): (2.21)The (weighted) set C = (C;m) is a (weighted) � -design in Sn�1 with respect tothe substitution �(d) = 1 � d2=2 if and only if the approximation formula (2.21)becomes equality for all functions u(x) which are polynomials in coordinates of x =(x1; :::; xn) 2 Sn�1 of degree at most �: Thus B(Sn�1; � +1) is the minimum numberof nodes in the approximation formula of the sort.Example 2.4. Real, complex, and quaternionic projective spaces. For m = 1; 2; and4; consider Tnm = fu = (u1; :::; un); ui 2 Tmg ,where T1 = R is the real number �eld, T2 = C is the complex number �eld, andT4 = H is the associative (but noncommutative) quaternionic algebra. In H there isthe basis 1 (the unity), i; j; k, such that i2 = j2 = k2 = �1; ij = �ji = k; and anyu 2 H can be represented in the form u = x0+x1i+x2j+x3k, where x0; x1; x2; x3 2 R.ThusR � C �H andm is the dimension of Tm overR. For any u 2 H the conjugateelement u� = x0�x1i�x2j�x3k is de�ned. Since uu� = u�u = x20+x21+x22+x23, onecan de�ne a norm juj of u 2 Tm by means of juj = puu� and verify that (uv)� = v�u�and juvj = juj�jvj for any u; v 2 Tm. For any vectors (or points) u = (u1; :::; un) 2 Tnmand v = (v1; :::; vn) 2 Tnm, de�ne the inner product as follows:(u;v) = nXi=1 uiv�i : (2.22)Points u = (u1; :::; un) 2 Tnm and v = (v1; :::; vn) 2 Tnm are said to be equivalent ifthere exists a non-zero � 2 Tm such that vi = �ui for each i = 1:::; n. Elements ofthe projective spaces TmPn�1 are de�ned as equivalence classes of non-zero points ofTnm and called lines (through the origin in Tnm). For any U; V 2 TmPn�1, one cande�ne the angle ' = '(U; V ), 0 � ' � �2 , between lines U and V by means ofcos'(U; V ) = j(u;v)jp(u;u)(v;v) for any u 2 U; v 2 V , (2.23)since the right-hand side of (2.23) does not depend on the choice of points u;v onthese lines, and the distanced(U; V ) =p1� cos'(U; V ) = p2 sin '(U; V )2 . (2.24)For the metric space X = TmPn�1 we have �(X) = [0; 1] and D(X) = 1. Thefollowing facts are well-known (see [42], [59], [53]). The isometry group of TmPn�110

for m = 1; 2; and 4, respectively, consists of all orthogonal, unitary, and symplecticmatrices of order n and acts transitively on TmPn�1. As in the case of Sn�1, onTmPn�1 there exists a unique normalized invariant measure �. The metric spaceTmPn�1 is distance invariant and for any d = p2 sin '2 , 0 � ' � �2 ;�(d) = �(m2 n)�(m2 (n� 1))�(m2 ) 1Zcos2 ' (1� z)m2 (n�1)�1zm2 �1dz. (2.25)Example 2.5. Graphs as metric spaces. Consider an arbitrary connected undirectedgraph � (without loops and multiple edges). The set X = X(�) of vertices of �may be considered as a �nite metric space with the path metric d(x; y) equal to theminimum number of edges in a path connecting the vertices x and y of �: In this case�(X) = f0; 1; :::; D(X)g and D(X) coincides with the diameter of �: In particular, ifwe consider complete, cyclic and linear graphs on a vertex set f0; 1; :::; v � 1g; v � 2;we obtain respectively metric spaces X with distancesd(x; y) = 1� �x;y ;d(x; y) = min(jx � yj; v � jx� yj); (2.26)d(x; y) = jx� yj; (2.27)and D(X) = 1; D(X) = b v2 c; D(X) = v�1: The �rst two of these spaces are distanceinvariant; the third is not for v � 3: (It should be noted that the concept of a distanceinvariant metric space is stronger than the usual concept of a regular graph.) Noticethat the value A(X(�); 2) is the maximum size of a set of non-pairwise adjacentvertices (independent set) of �:Example 2.6. Degrees of metric spaces. For a metric space X = (X; d(x; y)) weconsider a space X(n) consisting of vectors x = (x1; :::; xn) where xi 2 X; i = 1; :::; n:We can determine a distance in X(n) in di�erent ways, for example, asd(x;y) = nXi=1 d(xi; yi) (2.28)or d(x;y) = max1�i�n d(xi; yi): (2.29)Notice that if X = X(�) for some graph �; then the set X(n) of vertices and a set ofedges E such that (x;y) 2 E if and only if d(x;y) = 1 form a graph �(n) = (X(n); E)with path metric either (2.28) or (2.29). In particular, if we use this construction withthe digit-by-digit metric (2.28) in complete and cyclic graphs on v vertices, we obtainthe Hamming and Lee metrics and corresponding spaces. For an arbitrary connectedgraph � on v vertices the value npA(X(�(n)); 2); where the path metric (2.29) isused, has a limit as n!1 and this limit is called the Shannon capacity [101] of �:11

Analogous constructions can also be used in the in�nite case. In particular, weconsider a metric space X = (I; d(x; y); �) with I = [0; 1) and distance (2.26), wherev = 1, and with I = [0; 1] and distance (2.27) (� is the normalized Lebesgue measureon I in the both cases). The degree X(n) with distance (2.29) and with measureequal to the normalized n-dimensional volume gives rise to the n-dimensional torusTn = ([0; 1)n; d(x;y)) with metricd(x;y) = max1�i�n min(jxi � yij; 1� jxi � yij) (2.30)and to the n-dimensional cube In = ([0; 1]n; d(x;y)) with metricd(x;y) = max1�i�n jxi � yij: (2.31)It is clear that Tn is a distance invariant space while In is not, and D(Tn) = 12 ;D(In) = 1:2.2. A system Q of orthogonal polynomials for a compact metric spaceWe endow a compact metric space X = (X; d(x; y); �) and a given substitution �(d)with a system of orthogonal polynomials Qi(t) of degree i; i = 0; 1; :::; s(X): Thissystem Q = fQi(t)g is in fact de�ned by the substitution �(d) and the mean measure�(d) of closed metric balls of radius d: Considering some properties of the system Q weobtain universal bounds on the sizes of codes and designs in compact metric spaces.Notice that the de�nition of a (weighted) � -design (with respect to substitution �(d))is invariant under any linear transformation of �(d): This allows us to assume withoutadditional loss of generality that �(d) is a continuous strictly decreasing function on[0; D(X)] such that �(D(X)) = �1 � �(d) � �(0) = 1: (2.32)Such a substitution function �(d) is referred to as standard. The inverse function ofthe standard function �(d) is denoted by ��1( ); i.e., for any t; �1 � t � 1; ��1(t) = dwhen �(d) = t:Consider the following inner product for continuous functions in a real t on theinterval [�1; 1] : f � g = ZX ZX f(�(d(x; y)))g(�(d(x; y)))d�(x)d�(y): (2.33)We verify that the integral on the right-hand side of (2.33) can be rewritten as theRimann-Stieltjes integral on [�1; 1]: To do this we consider the function �(t) in a realt on the interval [�1; 1] de�ned by�(t) = 1� �(��1(t)); (2.34)12

which increases with t from �(�1) = 0 up to �(1) = 1� �(0): When X is �nite and�(X) = fd0 = 0; d1; :::; dsg where s = s(X); �(t) is left continuous (i.e., �(t � 0) =�(t)) and has s+ 1 steps at the points ti = �(di) with positive step sizeswi = �(ti + 0)� �(ti) = �(di)� �(di � 0) = 1jX j2 Xx2XBdi(x;X); (2.35)i = 0; 1; :::; s; sPi=0wi = 1: In the in�nite case we assumed that �0�(x; �) is continuous asa function in two variables � 2 [0; D(X)], x 2 X and hence �0(�) = RX �0�(x; �)d�(x).Therefore, for some natural additional assumptions on �(d), �(t) is di�erentiableand w(t) = �0(t) = ��0(d)�0(d) is continuous on [�1; 1] and positive inside the interval.An arbitrary left continuous function �(t) on [�1; 1] generates the Lebesgue-Stieltjesmeasure � on the same interval by�[a; b] = �(b+ 0)� �(a); �(a; b) = �(b)� �(a+ 0); (2.36)�(a; b] = �(b+ 0)� �(a+ 0); �[a; b) = �(b)� �(a): (2.37)In our case the measure � is normalized (�[�1; 1] = 1),�[a; b] = �(da)� �(db � 0) where a = �(da); b = �(db);and under the above assumptions, we can rewrite (2.33) asf � g = 1Z�1 f(t)g(t)d�(t): (2.38)Since j�0(X)j = s(X); the polynomials ti; i = 0; :::; s(X); for t = �(d); d 2 �(X); arelinearly independent, and using them in the orthogonalization process with respectto (2.38) we obtain the following statement.Theorem 2.7. For a compact metric space X = (X; d(x; y); �) with a �xed standardsubstitution �(d) there exists a unique system Q of orthogonal polynomials Qi(t) ofdegree i on the interval [�1; 1] and a unique system of positive constants ri (i =0; 1; :::; s(X)) such that ri 1Z�1 Qi(t)Qj(t)d�(t) = �i;j ; (2.39)Qi(1) = 1; i = 0; 1; :::; s(X); (2.40)where �(t) is de�ned by (2.34). 13

Notice that Q0(t) � 1 and r0 = 1 (2.41)since the measure is normalized. Under our assumption, the orthogonality condition(2.39) for a �nite and in�nite X takes respectively the following form:ri sXk=0Qi(�(dk))Qj(�(dk))wk = �i;j ; i; j = 0; 1; :::; s = s(X); (2.42)ri 1Z�1 Qi(t)Qj(t)w(t)dt = �i;j ; i; j = 0; 1; ::: : (2.43)Example 2.8. For the Hamming space X = Hnv with the standard substitution�(d) = 1� 2dn we have s = n; di = i; wi = v�n�ni�(v � 1)i, i = 0; 1; :::; n: Conditions(2.39) and (2.40) uniquely determine constants ri = wivn and polynomials Qi(t)which are connected with the Krawtchouk polynomials [61]Kn;vi (z) = iXj=0(�1)j(v � 1)i�j�zj��n� zi� j� (2.44)as follows Qi(t) = (ri)�1Kn;vi (n(1� t)2 ): (2.45)Example 2.9. For the Johnson space Jnw (w � n=2) with the standard substitution�(d) = 1� 2dw we have s = w; di = i; wi = �wi ��n�wi �=�nw�, i = 0; 1; :::; w: Conditions(2.39) and (2.40) uniquely determine constants ri = �ni��� ni�1� and polynomials Qi(t)which are connected with the Hahn polynomialsJi(z) = iXj=0(�1)j �ij��n+1�ij ��wj ��n�wj � �zj� (2.46)as follows Qi(t) = Ji(w(1� t)2 ): (2.47)Example 2.10. For the Euclidean sphere X = Sn�1 with the standard substitution�(d) = 1� d22 we have s = 1 and w(t) = �0(t) = �(n2 )�(n�12 )�( 12 )(1 � t2)n�32 (see (2.34),(2.19) and (2.20)). It is known [14] that the Jacobi polynomialsP�;�i (t) = 12i�i+�� � iXj=0�i+ �j ��i+ �i� j�(t� 1)i�j(t+ 1)j ; (2.48)14

(� � � 12 ; � � � 12 ), normalized by P�;�i (1) = 1; satisfy the following orthogonalitycondition:2i+ �+ � + 1i+ �+ � + 1 �i+ �+ � + 1i ��i+�i ��i+�i � c�;� 1Z�1 P�;�i (t)P�;�j (t)(1� t)�(1 + t)�dt = �i;j ;(2.49)where the constant c�;� = �(�+ � + 2)2�+�+1�(�+ 1)�(� + 1) (2.50)normalizes the measure, i.e., c�;� 1R�1(1 � t)�(1 + t)�d(t) = 1: Therefore, in the caseSn�1; n = 2; 3; ::: , we have ri = 2i+n�2i+n�2 �i+n�2i � andQi(t) = P n�32 ;n�32i (t): (2.51)Example 2.11. For the projective spaces X = TmPn�1 (see Example 2.4) with thestandard substitution�(d) = 2 �1� d2�2 � 1 (or �(d) = cos 2' if d = p2 sin '2 ), (2.52)we have s = 1 and w(t) = �0(t) = �(m2 n)2m2 n�1�(m2 (n�1))�(m2 ) (1 � t)m2 (n�1)�1 (1 + t)m2 �1using (2.34), (2.25), and the change of variable 2z = t + 1. Therefore, in the caseTmPn�1; m = 1; 2; and 4; n = 3; 4; ::: , with this substitution,Qi(t) = P m2 (n�1)�1;m2 �1i (t): (2.53)Example 2.12. In the case of the n-dimensional torus Tn (see Example 2.6) wehave �(x; �) = (2�)n for any x 2 Tn and any �; 0 � � � D(Tn) = 12 : The linearsubstitution �(d) = 1 � 4d is standard and by (2.34) �(t) = 1 � �1�t2 �n and hencew(t) = �0(t) = n2�n(1 � t)n�1: According to Theorem 2.7 and (2.49) in this caseQi(t) coincides with the Jacobi polynomials Pn�1;0i (t) and ri = n+2in �n�1+ii �2: Forthe n-dimensional cube In; �(x;�) in general depends on x 2 In: However it is easyto calculate that �(�) = RIn �(x;�)d�(x) = (2� � �2)n for any �; 0 � � � D(In) = 1:The substitution �(d) = 1 � 2(2d � d2) is standard and by (2.34) �(t) also equals1� �1�t2 �n: Hence the polynomials Qi(t) and constants ri coincide with those for thetorus Tn but under the latter standard substitution.2.3. A restricted �-design problem for systems of orthogonal polynomialsWe formulate an extremum problem for a system of orthogonal polynomials. Thisproblem has a unique optimal solution. An application to the system Q considered15

above gives a universal bound on the size of weighted � -designs in a compact metricspace X .Let F [t] be the set of polynomials in a real t with real coe�cients. Any f 2 F [t]of degree at most s = s(X) can be uniquely represented in the form sPi=0 fiQi(t) wherefi = fi(Q) = ri 1Z�1 f(t)Qi(t)d�(t): (2.54)We also introduce the notation(f) = Q(f) = f(1)f0 when f0 6= 0: (2.55)Remark 2.13. We can extend de�nitions (2.54) and (2.55) to polynomials f(t) ofdegree more than s when s = s(X) (and X) are �nite. It is correct because in thiscase, the polynomials f(t) and sPi=0 fiQi(t); where fi are de�ned in (2.54), coincide atall t = �(d) when d 2 �(X):In the sequel, we bear in mind that f0(Q) > 0 for any nonzero polynomial f 2 F [t](of degree at most s(X) in the case of �nite X) such that f(t) � 0 for �1 � t � 1:Notice that by (2.33), (2.38)-(2.41) and (2.54)f0 = ZX ZX f(�(d(x; y)))d�(x)d�(y) (2.56)and f0 is equal to zero when f(t) = Qi(t); i = 1; :::; s(X). This gives the followingcorollary.Corollary 2.14. A weighted set C = (C;m) is a weighted � -design if and only ifXx2CXy2CQi(�(d(x; y)))m(x)m(y) = 0; i = 1; :::; �: (2.57)Consider an arbitrary weighted set C = (C;m). A polynomial f 2 F [t] is calledannihilating for C if f(�(d(x; y))) = 0 for any x; y 2 C; x 6= y: A polynomial,annihilating for C; of minimal degree (i.e., s(C)) is called minimal. Let fC(t) denotethe minimal polynomial for C such that fC(1) = 1: Let�0(C) = fi : i 2 f0; 1; :::; s(X)g; B0i(C) 6= 0g: (2.58)where B0i(C) = rijCj Xx;y2CQi(�(d(x; y)))m(x)m(y); i = 0; 1; :::; s(X) (2.59)16

(the normalization (2.11) is used). Notice thatB00(C) = jCj (2.60)and hence 0 2 �0(C): Let �00(C) = �0(C)nf0g: From the de�nition (2.13) of the dualdistance d0(C) and Corollary 2.14 it follows (cf. (2.6)) thatd0(C) = min�00(C) (2.61)when the set �00(C) is non-empty. For codes C in a �nite X we consider also twoother parameters (cf. (2.5) and (2.7)): the number of (nonzero) dual distancess0(C) = j�00(C)j (2.62)and �0(C) = � 1 if s(X) 2 �00(C);0 otherwise. (2.63)For an arbitrary weighted set C = (C;m) and an arbitrary f 2 F [t] by (2.59) wehave Xx;y2C f(�(d(x; y)))m(x)m(y) = jCj s(X)Xi=0 fi(Q)ri B0i(C): (2.64)Theorem 2.15. For any weighted � -design C = (C;m) and any polynomial f 2 F [t]of degree at most � such that f0(Q) > 0 and f(t) � 0 for �1 � t � 1;jCj � Q(f) (2.65)with equality if and only if C is a simple (i.e., m(x) � 1 for all x 2 C) � -design andf(t) is annihilating for C.Proof. By Corollary 2.14 and (2.59), B0i(C) = 0 for i = 1; :::; � and hence (2.64),(2.60), (2.32) and (2.11) imply the following inequalitiesf0 � f(�(0)) Px2Cm2(x)� Px2Cm(x)�2 � f(1)jCjwith the above mentioned conditions of attainability.This theorem gives rise to the following extremum problem for the system Q oforthogonal polynomials whose solution ensures the best bound in (2.65).A restricted1 � -design problem: In the class of polynomials f 2 F [t] of degreeat most � such that f(t) � 0 for �1 � t � 1 (and hence f0(Q) > 0) �nd a polynomialwhich maximizes Q(f):1The term "restricted" means that we only consider polynomials of degree at most � ; later weremove the restriction for polynomial metric spaces.17

In Section 5 we prove (as a consequence of a result in the theory of orthogonalpolynomials) that there exists a unique (up to a constant factor) solution f = g� ofthe problem. To describe g� and calculate Q(g� ) we de�ne, for any a; b 2 f0; 1; ::: g;systems Qa;b of orthogonal polynomials which in a certain sense are adjacent to anarbitrary system Q satisfying the conditions (2.39) and (2.40). As before we assumethat �(t) is a left continuous step function or continuously di�erentiable, and hence(2.42) or (2.43) holds. First we choose a function �a;b(t) and constants ca;b as follows.If �(t) is continuously di�erentiable and �0(t) = w(t); then so is �a;b(t) and (�a;b(t))0 =ca;b(1� t)a(1 + t)bw(t): If �(t) is a step function, then so is �a;b(t_) and it is producedfrom �(t) by multiplying its steps wi at the points ti = �(di) (i = 0; 1; :::; s) by ca;b(1��(di))a(1+�(di))b (thus the number sa;b+1 of steps of �a;b(t) equals s�1+�a;0+�b;0since �(d0) = 1 and �(ds) = �1; we put sa;b =1 in the continuous case). Constantsca;b are chosen so that the Lebesgue-Stieltjes measure �a;b on [�1; 1] generated by thefunction �a;b(t) (similar to (2.36)-(2.37)) is also normalized, i.e.,1Z�1 d�a;b(t) = ca;b 1Z�1 (1� t)a(1 + t)bd�(t) = 1: (2.66)Similar arguments as in Theorem 2.7 show that there exists a unique system Qa;b oforthogonal polynomials Qa;bi (t) of degree i on the interval [�1; 1] and a unique systemof positive constants ra;bi (i = 0; 1; :::; sa;b) such thatra;bi 1Z�1 Qa;bi (t)Qa;bj (t)d�a;b(t) = �i;j ; (2.67)Qa;bi (1) = 1; i = 0; 1; :::; sa;b: (2.68)It is easily seen that Qa;b0 (t) = 1; ra;b0 = 1; Q0;0i (t) = Qi(t); r0;0i = ri; c0;0 = 1:Sometimes we shall omit the indices a; b if they both equal 0. LetT a;bi (x; y) = iXj=0 ra;bj Qa;bj (x)Qa;bj (y); i = 0; 1; :::; sa;b: (2.69)We shall see (Lemma 5.24 and Remark 5.27) thatQa;b+1i (t) = T a;bi (t;�1)T a;bi (1;�1) ; Qa+1;bi (t) = T a;bi (t; 1)T a;bi (1; 1) : (2.70)We let ta;bi denote the largest root of polynomial Qa;bi (t) and putda;bi = ��1(ta;bi ): (2.71)18

To describe the optimal polynomial for the restricted � -design problem we repre-sent � as an odd or even integer in the form� = 2l� � where l = l(�) 2 f1; 2; ::: g and � = �(�) 2 f0; 1g :It is clear that l = b �+12 c and � = 2b �+12 c � �: Letg(�)(t) = (1 + t)�0@ l��Xj=0 r0;�j Q0;�j (t)1A2 : (2.72)Note that l��Xj=0 r0;�j Q0;�j (t) = T 0;�l��(1; 1)Q1;�l��(t) = Q1;�l��(t) l��Xj=0 r0;�j : (2.73)In accord with (2.67)r0;�i c0;� 1Z�1 Q0;�i (t)Q0;�j (t)(1 + t)�d�(t) = �i;j ;and, hence, by (2.54) and (2.66)g(�)0 (Q) = 1Z�1 g(�)(t)d�(t) = (c0;�)�1 l��Xi=0 r0;�i ;where c0;0 = 1 and (c0;1)�1 = 1Z�1 (1 + t)d�(t) = �2Q1(�1)1�Q1(�1) ; (2.74)since 1 + t = 2Q1(t)�Q1(�1)1�Q1(�1) : Using the polynomial g(�) in Theorem 2.15 and takinginto account (2.68)-(2.70), we get the following theorem.Theorem 2.16. For any weighted � -design C = (C;m) in a compact metric spaceX; jCj � Q(g(�)) = �1� 1Q1(�1)�� l��Xi=0 r0;�i ; (2.75)where l = b �+12 c and � = 2l � � with equality in (2.75) holding if and only if C is asimple � -design and (1 + t)�Q1;�l��(t) is annihilating for C:A � -design C for which the bound (2.75) is attained is called a tight design.19

Lemma 2.17. For any code C in a compact metric space X�(C) � 2s(C)� �(C): (2.76)Proof. For any code C there exists a polynomial g(t) of degree s�� where s = s(C)and � = �(C); such that the polynomial f(t) = (1� t)(1 + t)�(g(t))2 is annihilatingfor C and, moreover, the left-hand side of (2.64) equals zero. The polynomial f(t) isnonnegative for �1 � t � 1 and by (2.54), f0(Q) > 0 (if its degree 2s � � + 1 doesnot exceed s(X) in the case of �nite spaces X). Therefore, the inequalitiess(X) � �(C) � 2s(C)� �(C) + 1imply that B0i(C) = 0; i = 1; :::; 2s� � + 1; and contradict the equality (2.64) for thepolynomial f(t) of degree 2s� � + 1 � s(X):Lemma 2.18. A code C in a compact metric space X is a tight design if and only if�(C) = 2s(C)� �(C) and fC(t) = ( 1+t2 )�(C)Q1;�(C)s(C)��(C)(t):Proof. If the polynomial (1 + t)�Q1;�l��(t) where � 2 f0; 1g is annihilating for a codeC; then s(C) � �(C) � l � �; and �(C) � � since �(D(X)) = �1 and Q1;0l (�1) 6= 0:Therefore for a tight (2l� �)-design C,�(C) � 2l � � � 2s(C)� 2�(C) + � � 2s(C)� �(C)and hence by Theorem 2.16 and Lemma 2.17 �(C) = 2s(C) � �(C); l = s(C); � =�(C): On the other hand, if the conditions of the lemma are satis�ed, then usingf(t) = fC(t) in (2.64) shows that jCj = (fC): However, taking into account (2.68)-(2.70) again it is easy to verify that (fC) = (g(�(C))); and we have equality in(2.75).The optimality and uniqueness of the polynomial g(�) for the restricted � -designproblem will be proved in Section 5.We shall see that the systems Q0;1 for the Hamming spaceHnv and for the Johnsonspace Jnw coincide with systems Q for the spaces Hn�1v and Jn�1w�1 respectively. For theEuclidean sphere Sn�1, the projective spaces TmPn�1, the torus Tn, and the cubeIn, the polynomials Qi(t) are equal to Jacobi polynomials P�;�i (t) with some � and� ; hence, Q0;1i (t) = P�;�+1i (t) in these cases. It is known (see, for example, [14], [67])that for normalized Jacobi polynomials P�;�i (t) (see (2.48)-(2.49)),iXj=0 r�;�j = �i+ �+ � + 1i ��i+ �+ 1i �=�i+ �i � (2.77)and 1� 1P�;�1 (�1) = �+ � + 2� + 1 : (2.78)20

Therefore, for Qi(t) = P�;�i (t) the right-hand side of (2.75) equals�l + �+ � + 1l ��l + �+ 1� �l � � �=�l + �l � (2.79)We can use the examples considered and Theorem 2.16 to obtain the lower boundson the size of a weighted (2l � �)-design given in Table 2.1. In the connection withthese bounds for the projective spaces the following references are relevant: [30], [34],[35], [57], [82], [53], and [9].3. Polynomial metric spaces and extremum problems for thesystem Q of orthogonal polynomials3.1. Inequalities for nonnegative de�nite functionsAlmost all universal bounds for codes in metric spaces are based on inequalities forfunctions in two points x and y of the space which are nonnegative de�nite and de-pend only on the distance between x and y. The well known universal bounds ofBlichfeldt [17] and Rankin [91] are in fact connected with the usual inner product inEuclidean space. Important advances were made in the works of Sidelnikov [103] andWelch [123] which used inequalities for degrees of the inner product or of its modu-lus. The next step was taken by Delsarte [30] who described all nonnegative de�nitefunctions depending on the distance for so-called P - and Q-polynomial associationschemes by using a system Q of orthogonal polynomials (this system coincides withthat of Section 2.2). An analogous description is known (see, for example, [95], [121])for some in�nite metric spaces including the Euclidean sphere and real, complex andquaternionic projective spaces. This gives rise to some extremum problems for sys-tems of orthogonal polynomials which are linear programs in the case of �nite metricspaces. For the Hamming and Johnson spaces McEliece, Rodemich, Rumsey, Jr. andWelch [79] proposed a solution of the linear program which gives the best known as-ymptotic bound (MRRW-bound) when the code distance grows linearly with length.An analogous solution was also used for the case of the Euclidean sphere and projec-tive spaces by Kabatyanskii and the author [57] in order to obtain the best knownasymptotic bound (KL-bound) when the code distance or angle distance is a constantand the dimension tends to in�nity. We shall describe a better solution which wasfound by the author [64] and prove its optimality in a certain sense. This result givesthe MRRW- and KL-bounds as a corollary.For a metric space X = (X; d(x; y)) we discuss real- and complex-valued functionsF (x; y) in two variables x; y 2 X . In the case of a �niteX , F (x; y) might be consideredas an element of a matrix of order jX j � jX j in row x 2 X and column y 2 X . Afunction F (x; y) is called Hermitian if F (x; y) = F (y; x) for any x; y 2 X where a isthe conjugate of a.A Hermitian function F (x; y) is called nonnegative de�nite on X if for any �nite21

Space Standard substitution Lower bound on ReferencesX �(d) B(X; 2l� � + 1)Hnv 1� 2dn v� l��Pj=0 �n��j �(v � 1)j [92]Jnw 1� 2dw �nw���n��l�� � [93] when � = 0[38] when � = 1Sn�1 1� d22 �n+l�2l�1 �+ �n+l�1��l�� � [35]RPn�1 2 �1� d2�2 � 1 �n+2l�1��2l�� �CPn�1 2 �1� d2�2 � 1 �n+l�1l ��n+l�1��l�� �HPn�1 2 �1� d2�2 � 1 1l+1�2n+l�1l ��2n+l�2��l�� �Tn 1� 4d �n+ll ��n+l��l�� �In 1� 2d(2� d) �n+ll ��n+l��l�� �Table 2.1: Lower Bounds on the Size of (2l � �)-Designs22

set (code) C � X and any complex function v(x); x 2 X :Xx;y2C F (x; y)v(x)v(y) � 0 (3.1)(since F (x; y) is Hermitian, the left-hand side of (3.1) is real). For a �nite metricspace X; a matrix F (x; y); x; y 2 X; is nonnegative de�nite if and only if there existsome number h of functions w1(x); :::;wh(x) on X such thatF (x; y) = hXj=1wj(x)wj(y): (3.2)For our purposes in the case of in�nite metric spaces it is also su�cient to consideronly nonnegative de�nite functions of this type. We shall call a function which canbe represented in the form (3.2) with continuous nonzero functions wj(x); j = 1; :::; h(h � 1); a �nite dimensional2 nonnegative de�nite function (FDNDF) on X . It issigni�cant to note that for any code C � X and any FDNDF F (x; y) on X we haveF (C) = 1jCj2 Xx;y2C F (x; y) = 1jCj2 hXj=1 jXx2C wj(x)j2 � 0. (3.3)Example 3.1. The most signi�cant example of an FDNDF on the Euclidean complexspace Cn is the usual inner product of vectors x = (x1; :::; xn) and y = (y1; :::; yn)(x; y) = nXi=1 xiyi: (3.4)The function j(x; y)j is not an FDNDF on Cn but j(x; y)j2 is because ofj(x; y)j2 = nXi=1 nXj=1 xixjyiyj (3.5)The function Kn;vi (d(x; y)); where Kn;vi (z) is the Krawtchouk polynomial (2.44) ofdegree i and d(x; y) is the Hamming distance, can be represented in the formKn;vi (d(x; y)) = Xa2Hnv , d(0;a)=i �(a;x�y) (3.6)where � is a primitive v-th root of unity. Thus (3.2) holds with wj(x) = �(a;x) andh = �ni�(v � 1)i; and Kn;vi (d(x; y)) is an FDNDF3 on the Hamming space Hnv :2Later we consider the function F (x; y) as the kernel of a linear �nite dimensional operator onL2(X;�).3Of course, a real FDNDF F (x; y) also has a representation (3.2) with real functions; however,this example shows that the complex representation might be more useful.23

Notice the following properties of FDNDFs:i) A linear combination of FDNDFs with positive coe�cients is an FDNDF.ii) A product of FDNDFs is an FDNDF.iii) If F (x; y) is an FDNDF on X and � is a (continuous) mapping of a space Yinto X; then H(x; y) = F (�(x); �(y)) is an FDNDF on Y:Example 3.2. It is known that for any v (v = 2; 3; ::: ) on Sv�2 there exists anequidistant code of v points �(0); �(1); :::; �(v�1) with Euclidean distanceq 2vv�1 (thevertices of the regular simplex inscribed in Sv�2): The mapping � : Hnv ! S(v�1)n�1such that �(x) = 1pn (�(x1); :::; �(xn)) for x = (x1; :::; xn) 2 Hnv has the property that1� vd(x; y)(v � 1)n = (�(x); �(y)) (3.7)and hence this function is an FDNDF on Hnv : In the case of the Johnson space Jnw onecan map [104] any w-subset x of the n-set f0; 1; :::; n�1g to �(x) = (z1; :::; zn) 2 Sn�1as follows: zi = 8>><>>: �qn�wwn if i 2 x;q w(n�w)n otherwise.It follows that the function 1� nd(x; y)w(n� w) = (�(x); �(y)) (3.8)is an FDNDF on Jnw:To discuss simultaneously FDNDFs on �nite and compact in�nite metric spacesX with normalized measure � we consider the (possibly, �nite dimensional) Hilbertspace L2(X;�) of complex-valued functions u on X satisfyingkuk 2 = ZX ju(x)j2d� (x) <1with the inner product hu;vi = ZX u(x)v(x)d� (x) (3.9)(Functions are equivalent if they di�er on a set A with �(A) = 0, and hence equivalentcontinuous functions coincide by our assumption (2.3).) We associate with everyFDNDF F (x; y) a linear operator F which transforms u 2 L2(X;�) to the function24

Fu =ZX F (x; y)u (y) d� (y) = hXj=1wj (x) ZX wj(y)u(y)d� (y) (3.10)of a subspace V (F ) of continuous functions generated by w1 (x) ; : : : ;wh (x). This�nite dimensional operator F is self-conjugate, sincehFu;vi =hu;Fvi = hXj=1hu;wjihv;wji: (3.11)From (3.10) and (3.11) it follows that all eigenvalues of F are nonnegative, all eigen-functions with positive eigenvalues belong to V (F ) ; and all eigenfunctions with a zeroeigenvalue are orthogonal to V (F ) : Hence, by the Hilbert-Schmidt Theorem (see, forexample [58]) we have the following result.Lemma 3.3. For any FDNDF F (x; y) on X; in V (F ) there exists an orthonor-mal basis of m = dimV (F ) continuous eigenfunctions e1(x); : : : ; em(x) with positiveeigenvalues �1; : : : ; �m (i.e., hei; eji = �ij , Fei=�iei) such that any continuous func-tion u(x) on X can be uniquely represented asu(x) = mXi=1 uiei(x) + eu(x) (3.12)where ui= hu; eii, heu; eii = 0; i = 1; :::;m; Feu = 0, and (3.2) is reduced to thefollowing diagonal form: F (x; y) = mXi=1 �iei(x)ei(y): (3.13)We should only add that (3.2) can be represented in the form (3.13) since wj(x) =Pmk=1hwj ; ekiek(x) by (3.12) andPhj=1hei;wjihek;wji = �i�i;k for i; k = 1; :::;m; by(3.11).Now we discuss how an FDNDF F (x; y) can be used for obtaining a (universal)upper bound on the size of a code C � X with minimal distance d (C) � d. We knowthat (3.3) is true. LetF (X) = ZX ZX F (x; y) d� (x) d� (y) = hXj=1 ������ZX wj(x)d� (x)������2 : (3.14)We shall say that for a given FDNDF F (x; y) the inequality on the mean holds ifF (C) � F (X) for any �nite C � X: (3.15)25

It is not true that the inequality on the mean holds for any FDNDF. Therefore it isnatural for an FDNDF F (x; y) to �nd the maximum nonnegative � = � (F ) such thatF (x; y) � �J (x; y) is FDNDF (here as usual J (x; y) = j(x)j (y) where j(x) � 1 forx 2 X). Since J (C) = J (X) = 1 (see (3.3),(3.14)) we have F (C) � � (F ), F (X) �� (F ) and that for F (x; y) the inequality on the mean holds when � (F ) = F (X) :Example 3.4. One can check that for any x; y 2 Cn;j(x; y)j2 � 1n nXi=1 jxij2 nXi=1 jyij2 =Xi 6=j xixjyiyj + 12nXi 6=j �jxij2 � jxj j2��jyij2 � jyj j2� :It follows that for the FDNDF F (x; y) = j(x; y)j2 (see (3.5)) considered on the unitEuclidean sphere Sn�1 (and on the unit complex Euclidean sphere CSn�1) we have�(F ) � 1n : The integral (3.14) equals (see Example 2.10)1Z�1 t2w(t)dt = �(n2 )�(n�12 )�( 12 ) 1Z�1 t2(1� t2)n�32 dt = 1n:(We used the following equality1Z0 za�1(1� z)b�1dz = �(a)�(b)�(a+ b) , (3.16)for the beta-function.) Therefore F (X) = 1n and hence �(F ) = F (X) = 1n (this isalso true for CSn�1).The value � (F ) for a symmetric real matrix F (x; y) was calculated in [80]. Weuse it and �nd necessary and su�cient conditions for �(F ) = F (X):Lemma 3.5. Let an FDNDF F (x; y) be reduced to the form (3.13) and letj(x) = mXi=1 ai ei(x) + ej(x); (3.17)where m = dim V (F ) � 1 and Fej = 0 . If j(x) =2 V (F ), then � (F ) = 0 � F (X)with equality if and only if a1 = : : : = am = 0 ; if j(x) 2 V (F ) ; then� (F ) = mXi=1 jaij2�i !�1 � F (X) (3.18)with equality if and only if all �i corresponding to nonzero ai are equal (and equal to� (F )). 26

Proof. Notice that F (X) = hF j; ji = mXi=1 �i jaij2 (3.19)and if F (x; y)��J (x; y) is an FDNDF then for any function u (see (3.12)) such thathu; j i = 1 we have hFu;ui = mXi=1 �i juij2 � �: (3.20)In the case when j(x) =2 V (F ) the continuous function ej(x) is not identically zero,and we have kejk 6= 0. Since hej; j i = kejk2, for u = ej= kejk2 we have hu; j i = 1and Fu = 0. By (3.19) and (3.20) the �rst part of the lemma is proved. In the casewhen j(x) 2V (F ), hj; ji = mXi=1 jaij2 = 1 (3.21)and at least one ai is not zero. By the Cauchy inequality����� mXi=1 uiai�����2 � mXi=1 �i juij2 � mXi=1 jaij2�i (3.22)with equality if and only if the vectors (u1; : : : ; um) and � a1�1 ; : : : ; am�m� are propor-tional. Let �0 = mXi=1 jaij2�i !�1 .For ui = �0�i ai; i = 1; : : : ;m; we have jPmi=1 uiaij = 1 and equality in (3.22). Therefore(3.20) implies that � (F ) � �0. On the other hand, as j (x) 2 V (F ), the functionmXi=1 �i �ei (x) � �0�i aij(x)��ei (y)� �0�i aij(y)�is an FDNDF and coincides with F (x; y) � �0J (x; y). Hence F (X) � � (F ) � �0.The last statement of the lemma follows from (3.21) and (3.19) and the necessary andsu�cient conditions of attainability of (3.22) for ui = ai; i = 1; : : : ;m.This proof shows that �(F ) = min hFu;u i; where the minimum is taken overall functions u such that hu,j i = 1: Furthermore, it can be extended to prove thefollowing statement for the space X(n) = X� :::�X (see [80]): if Fi(x; y); i = 1; :::; n;are FDNDFs on X; then F(x;y) = nYi=1Fi(xi; yi); (3.23)27

where x = (x1; :::; xn);y = (y1; :::; yn) 2 X(n) ; is an FDNDF on X(n) and�(F) = nYi=1�(Fi): (3.24)The general method of using an FDNDF to bound the size of a code is based onthe following.Lemma 3.6. For any FDNDF F (x; y) on X and any code C; C � X;maxx2C F (x; x) � maxx;y2Cd(x;y)�d(C) ReF (x; y) � jCj �(F )� maxx;y2Cd(x;y)�d(C) ReF (x; y)! ; (3.25)and, in particular, if maxx;y2C; d(x;y)�d(C) ReF (x; y) � 0; thenmaxx2C F (x; x) � jCj�(F ): (3.26)Proof. It is immediate from1jCj maxx2C F (x; x) + jCj � 1jCj maxx;y2Cd(x;y)�d(C) ReF (x; y) � 1jCj2 Xx;y2C F (x; y) = F (C) � �(F ):Remark 3.7. Lemma 3.6 gives a universal upper bound on the size of a code witha given minimal distance if the right-hand side of (3.25) or (3.26) is positive. Theproof shows that this bound can be improved replacing �(F ) by F (X) if for F (x; y)the inequality on the mean holds.Example 3.8. As was remarked in [80], [97], in particular, Lemma 3.6 gives theLov�asz bound [77] for Shannon capacity of a graph (see Example 2.6). Indeed, letM(x; y) be the (symmetric) adjacency matrix of a graph � with v vertices andN edgesand let �min and �max be the minimum and the maximum eigenvalues of M(x; y).Then �max > 0; �min < 0 andF (x; y) =M(x; y)� �min I(x; y); (3.27)where I(x; y) is the identity matrix, forms a FDNDF on X(�). Moreover, F (x; y)possesses the following properties:F (X(�)) = v�2(2N � v�min); (3.28)F (x; x) = ��min for any vertex x;and F (x; y) = 0 if d(x; y) � 2: (3.29)28

By (3.26) A(X(�); 2) � ��min�(F ) : Considering the space X(�(n)) = fx = (x1; :::; xn);xi 2 Xg with metric (2.29) and using the facts that the FDNDF (3.23) on X(�(n))(with Fi(x; y) = F (x; y)) has the propertyF(x;y) = 0 if d(x;y) � 2 (3.30)and (3.24) holds, we obtain the Lov�asz bound [77]:nqA(X(�(n)); 2) � ��min�(F ) : (3.31)In particular, the adjacency matrixM(x; y) of the cyclic graph with v; v � 3; verticescan be represented in the following diagonal form (cf. (3.13)):M(x; y) = v�1Xj=0 �jej(x)ej(y);where �j = v�1(�j + ��j); ej(x) = �jx and � = e 2�iv (it is easy to check using v�1Pj=0 �j =0). Since �j = 2v�1 cos 2�jv ; then �max = 2 and �min = �2 if v is even and �min =�2 cos �v if v is odd (the eigenvalues of the matrixM(x; y) are v times more than thoseof the operator M). Taking into account that F (x; y) = M(x; y) � �min I(x; y) =Pv�1j=0 (�j � v�1�min)ej(x)ej(y); j(x) = e0(x); �0 = 2v�1; we obtain by Lemma 3.5that �(F ) = v�1(2� �min) and, hence, for the cyclic graph � with v vertices,nqA(X(�(n)); 2) � v1� 2=�min : (3.32)Notice that the function j (x) � 1 is an eigenfunction of F if and only if F j =RX F (x; y) d�(y) does not depend on x 2 X . In this case the corresponding eigen-value is hF j; j i = F (X); however, it may happen that �(F ) < F (X):Nevertheless,we verify that the inequality on the mean (3.15) holds for any FDNDF F (x; y) ifRX F (x; y) d�(y) does not depend on x 2 X .To this end we consider some inequalities for an FDNDF F (x; y) on a �nite orin�nite set Y which depend on a choice of two measures, two subsets, and two func-tions on Y: The author found in [62] similar inequalities to explain the nature of theSidelnikov and Welch inequalities as those on the mean (3.15) and to obtain su�-cient conditions under which such inequalities are valid. We present here desirableinequalities in the general form published by Kabatyanskii and the author in [57].We will tacitly assume the measurability of all sets and the existence of all integralsconsidered below.Consider two normalized measures � and � on a �nite or in�nite set Y (�(Y ) = 1;�(Y ) = 1) and two complex functions g(x) and h(x). For an FDNDF F (x; y) on Y29

and any subsets A � Y and B � Y such that �(A) > 0; �(B) > 0; letF�;�(A;B; g; h) = 1�(A)�(B) ZA ZB F (x; y)g(x)h(y)d�(x)d�(y):Also let F�;�(A;B) = F�;�(A;B; 1; 1) and F�(A) = F�;�(A;A): In particular, for themetric space X with the measure �, F (X) = F�(X) = F�;�(X;X) and for any �niteset (code) C � X; F (C) = F�C (X) where the measure �C is de�ned as follows�C(A) = jA \ CjjCj for any A � X:Theorem 3.9. For any FDNDF F (x; y) on Y and any �; �; A; B; g and h;jF�;�(A;B; g; h)j2 � F�;�(A;A; g; g)F�;�(B;B; h; h): (3.33)To prove the theorem, twice change the order of summation, and use the Cauchyinequality for sequences and Fubini's Theorem (see [57]).Corollary 3.10. For an FDNDF F (x; y) on X the inequality on the mean holds ifRX F (x; y) d�(y) does not depend on x 2 X . In particular, for an FDNDF F (x; y) ona distance invariant space X the inequality on the mean holds if F (x; y) is a functionin d(x; y).Proof. The condition of the corollary implies that F�;�(A;X) = F�;�(X;X) = F�(X)for any measure � on X and any A � X: Therefore the inequality on the mean (3.15)follows from Theorem 3.9 for A = C; B = Y = X; � = �C ; � = �; g = h = 1: For adistance invariant space X , RX f(d(x; y))d�(y) does not depend on x 2 X because thefunction �(x; �) de�ned by (2.3) has the same property.Example 3.11 (The inequalities on the mean). Consider the Sidelnikov inequal-ity [103] for any code C � Sn�1:1jCj2 Xx;y2C(x; y)2s � �(n2 )�(s+ 12 )�(n2 + s)�( 12 ) ; s = 0; 1; ::: ; (3.34)and the Welch inequality [123] for any code C � CSn�1:1jCj2 Xx;y2C j(x; y)j2s � 1�n+s�1s � ; s = 0; 1; ::: , (3.35)which were proved by combinatorial methods. Both these important results can beproved as inequalities on the mean for FDNDFs. We already noticed that the inner30

product (x; y) = 1 � d2(x; y)=2 and, hence, F (x; y) = (x; y)2s; s = 0; 1; :::; is anFDNDF on Sn�1. UsingF (Sn�1) = ZSn�1 (x; y)2sd�(x) = �(n2 )�(n�12 )�( 12 ) 1Z�1 t2s(1� t2)n�32 dt(see Example 2.10) and (3.16) by Corollary 3.10 we get (3.34) and �nd that theright-hand side of (3.34) is equal to F (Sn�1): (F (Sn�1) = 0 for F (x; y) = (x; y)hif h is odd.) It was also proved [62] that (3.35) is the inequality on the mean forFDNDF F (x; y) = j(x; y)j2s on CSn�1 and the right-hand side of (3.35) is equal toF (CSn�1). (The measure on CSn�1 coincides with the measure � on S2n�1; if oneconsiders any x = (x1; :::; xn) 2 CSn�1 as (Re x1; Imx1; :::;Rexn; Imxn) 2 S2n�1:)Analogously using FDNDF functions (3.7) and (3.8) we have the following known[104] inequalities: for any code C � Hnv and any h = 0; 1; :::;1jCj2 Xx;y2C(1� vd(x; y)(v � 1)n)h � 1vn nXd=0(v � 1)d�nd�(1� vd(v � 1)n )h;and for any code C � Jnw and any h = 0; 1; :::;1jCj2 Xx;y2C(1� nd(x; y)w(n� w) )h � 1�nw� wXd=0�wd��n� wd �(1� ndw(n� w) )h:Consider also the function 'n;v;a;b(d) = (a� b)d(a+(v�1)b)n�d: From the de�nition(2.44) of Krawtchouk polynomials Kn;vi (z) it follows that for any d = 0; 1; :::; n;'n;v;a;b(d) = nXi=0Kn;vi (d)an�ibi:Since Kn;vi (d(x; y)) are FDNDFs on Hnv ; so is F (x; y) = 'n;v;a;b(d(x; y)) for anynonnegative a and b: For any p; 0 � p � v�1v ; a = 1v and b = 1v � pv�1 ; we have a � 0;b � 0 and hence by Corollary 3.10 the following inequality on the mean is valid: forany code C � Hnv and any p; 0 � p � v�1v ;1jCj2 Xx;y2C� pv � 1�d(x;y) (1� p)n�d(x;y) � F (Hnv ) = 1vn :This inequality turned out to be useful for studying the probability of undetectederrors over symmetric channels [75], [63]. Returning to Example 3.8 we can see thatfor F (x; y) de�ned by (3.27) the condition of Corollary 3.10 is ful�lled if and onlyif the graph � is regular with some valency v1. Then this condition is also ful-�lled for F(x;y) = nQi=1F (xi; yi). Moreover F(X(�(n))) = (F (X(�)))n and F(x;x) =31

(��min)n : According to (3.28), (3.30), Lemma 3.6 and Remark 3.7 it follows that forany regular graph � with v vertices and valency v1;nqA(X(�(n)); 2) � v1� v1=�min :In particular, this gives another proof of (3.32) for cyclic graphs (v1 = 2).Theorem 3.9 also gives rise to many useful inequalities for spaces which are notdistance invariant. For example, if we consider �nite sets A = fx1; :::; xMg � X ,B = fy1; :::; yNg � X; measures � = �A; � = �B and numbers g(xi) = gi; h(yj) = hj ;we obtain the following inequality for an FDNDF F (x; y) on an arbitrary X [62]:j MXi=1 NXj=1 F (xi; yj)gihj j2 � MXi=1 MXj=1 F (xi; xj)gigj NXi=1 NXj=1 F (yi; yj)hihj : (3.36)In particular, if X = Cn; N = 1; y = y1; g(xi) = 1; h(y) = 1 and F (x; y) = (x; y); wehave the inequality j MXi=1(xi; y)j2 � kyk2 MXi=1 MXj=1(xi; xj);which is equivalent to 2kyk2 MXi=1 MXj=1 d2(xi; xj) �4Mkyk2 MXi=1 d2(xi; y)� MXi=1 kxik2 �Mkyk2 � MXi=1 d2(xi; y)!2 :The last inequality implies the Blichfeldt inequality [17]MXi=1 MXj=1 d2(xi; xj) � 2M MXi=1 d2(xi; y);and for kyk = kxik = 1; i = 1; :::;M; gives the Rankin inequality [91]2 MXi=1 MXj=1 d2(xi; xj) � 4M � MXi=1 d2(xi; y)! MXi=1 d2(xi; y):If in (3.36) we put N = 1; y = y1; h(y) = 1 and use 2jgigj j � jgij2+jgj j2; we obtain forany FDNDF F (x; y) on an arbitrary X; any A = fx1; :::; xMg � X , and any y 2 X;the inequality j MXi=1 F (xi; y)gij2 � F (y; y) MXi=1 jgij2 MXj=1 jF (xi; xj)j: (3.37)32

In particular, if we consider in (3.37)gi = F (xi; y); gi = F (xi; y)jF (xi; y)j ; gi = F (xi; y)MPj=1 jF (xi; xj)j ;we obtain three inequalitiesMXi=1 jF (xi; y)j � F (y; y) max1�i�M MXj=1 jF (xi; xj)j; MXi=1 jF (xi; y)j!2 � F (y; y) MXj=1 MXj=1 jF (xi; xj)j;MXi=1 jF (xi; y)jMPj=1 jF (xi; xj)j � F (y; y);which in the case F (x; y) = (x; y) are due to Bombieri, Selberg and Hal�asz respectively(see [81]).Theorem 3.9 and values of known n-tuple integrals (see, for example, [40]) allowus to obtain di�erent generalizations of the Sidelnikov and Welch inequalities. Forexample, for any code C in an ellipsoid X = fx = (x1; :::; xn) 2 Rn : nPi=1�xiai�2 � 1gand any s = 0; 1; ::: ; the following inequality holds [62]:Xx;y2C(x; y)2s � �2ss �ls Xx2C nXi=1 x2i a2i!s!2 : (3.38)where ls = Pk1+:::+kn=s a4k11 � � � a4knn �2k1k1 � � ��2knkn � and the sum is taken over all orderedpartitions of s into n nonnegative integers (the generating function 1Pi=0 lizi is equal tonQi=1 �1� 4a4i z��1=2). This inequality follows from Theorem 3.9 when A = C; B = Y =X , � = �C and � = � where � is the normalized n-dimensional volume on the ellipsoidX . In particular, (3.38) implies that for any code C � X = fx = (x1; :::; xn) 2 Rn :nPi=1 x2i � 1g and any s = 0; 1; ::: ;Xx;y2C(x; y)2s � �(n2 )�(s+ 12 )�(n2 + s)�( 12 ) Xx2C nXi=1 x2i!s!2 ;that generalizes (3.34). 33

3.2. Polynomial metric spacesIn Section 2.2 for a compact metric space X = fX; d(x; y)g with a normalized mea-sure � and with a standard substitution �(d) (�(D(X)) = �1 � �(d) � �(0) =1) we de�ned by (2.39)-(2.40) the system of orthogonal polynomials Q = fQi(t);i = 0; 1; :::; s(X)g on the interval [�1; 1] and the system of positive constants ri;i = 0; 1; :::; s(X): Now we consider a class of compact metric spaces X for which thereexists a simple description of all FDNDFsF (x; y) = f(�(d(x; y))) (3.39)where f(t) is a polynomial in real t. Note that in the case of a �nite X , the polynomialQs(X)+1(t) = Yd2�(X)(t� �(d)) (3.40)of degree s(X)+1 has the property: Qs(X)+1(�(d(x; y))) = 0 for any x; y 2 X . Thus,without loss of generality, we can assume thatf(t) = s(X)Xi=0 fi(Q)Qi(t): (3.41)The function F (x; y) de�ned by (3.39) and (3.41) is a real-valued symmetric function,F (x; x) = f(1) for any x 2 X; and by (2.54) and (2.56)F (X) = ZX ZX f(�(d(x; y)))d�(x)d�(y) = 1Z�1 f(t)d�(t) = f0(Q): (3.42)It follows, from Corollary 3.10, that for any FDNDF F (x; y) of the form (3.39) on adistance invariant space X; the inequality of the mean F (C) � f0(Q) holds (for anyC � X).We shall call a compact metric space X FDNDF-polynomial if Qi(�(d(x; y))) is anFDNDF on X for any i = 0; 1; :::; s(X). For a FDNDF-polynomial space X (which,in general, is not distance invariant), let Fi(x; y) = Qi(�(d(x; y))); Vi = V (Fi); andmi = dimVi; and let Fi(x; y) have the following diagonal form (3.13):Fi(x; y) = Qi(�(d(x; y))) = miXj=1 �i;jei;j(x)ei;j(y): (3.43)The functions ei;j(x); i = 0; 1; :::; s(X); j = 1; :::;mi; form an orthonormal systemwith respect to the inner product (3.9), since by (2.39), (2.38) and (2.33),1Z�1 Qi(t)Qj(t)d�(t) = miXk=1 mjXl=1 �i;k�j;ljhei;k; ej;lij2 = 1ri �i;j : (3.44)34

Therefore, for the function F (x; y) de�ned by (3.39) and (3.41),Fei;j = fi(Q)�i;jei;j : (3.45)Notice that F0(x; y) = J(x; y); m0 = 1; e0;1(x) = j(x); �0;1=1 and hence by (3.45),F j = RX F (x; y)d�(y) equals f0(Q) and does not depend on x 2 X: This gives thefollowing theorem.Theorem 3.12. Let X be an FDNDF-polynomial space with the standard substitu-tion �(d). Then(i) the function F (x; y) de�ned by (3.39) and (3.41) is an FDNDF on X if andonly if fi(Q) � 0; i = 0; 1; :::; s(X);(ii) for any FDNDF F (x; y) de�ned by (3.39) and (3.41) the inequality on themean holds, and(iii) for any weighted set C = (C;m) and any i = 0; 1; :::; s(X);B0i(C) = rijCj Xx;y2CQi(�(d(x; y)))m(x)m(y)= rijCj miXj=1 �i;j jXx2C ei;j(x)m(x)j2 � 0: (3.46)Property (i) implies the following:Corollary 3.13. The systemQ of orthogonal polynomials for an FDNDF-polynomialspace X satis�es the following condition (Krein condition): for any i; k and h (0 �i; k; h � s(X)) there exist nonnegative (and positive when i + k = h) numbers qhi;ksuch that Qi(t)Qk(t) = s(X)Xh=0 qhi;kQh(t) (3.47)(for a �nite X; (3.47) is considered mod Qs(X)+1(t)).For any weighted set C = (C;m) and any polynomial (3.41) we have (2.64).Taking into account that f(�(d(x; y))) = f(1) if d(x; y) = 0 and �(d(x; y)) � �(d)if d(x; y) � d; and that B00(C) = jCj; B0i(C) � 0 for i = 1; :::; s(X); we obtain thefollowing re�nements of Lemma 3.6 and Theorem 2.15, respectively.Theorem 3.14. For any code C with minimal distance d(C) = d in an FDNDF-polynomial space X and any polynomial (3.41) such that f0(Q) > 0; fi(Q) � 0 fori = 1; :::; s(X) and f(t) � 0 for �1 � t � �(d);jCj � Q(f) = f(1)f0(Q) (3.48)with equality if and only if the polynomial f(t) is annihilating for C andfi(Q)B0i(C) = 0 for i = 1; :::; s(X): (3.49)35

Theorem 3.15. For any weighted � -design C = (C;m) in an FDNDF-polynomialspace X and any polynomial (3.41) such that f0(Q) > 0; fi(Q) � 0 for i = � +1; :::; s(X) and f(t) � 0 for �1 � t � 1;jCj � Q(f) = f(1)f0(Q) (3.50)with equality if and only if C is a simple � -design, the polynomial f(t) is annihilatingfor C and fi(Q)B0i(C) = 0 for i = � + 1; :::; s(X): (3.51)Note that by (2.40) and (3.43) for any x 2 X ,1 = Qi(�(d(x; x))) = miXj=1 �i;j jei;j(x)j2 (3.52)and hence miPj=1 �i;j = 1: (We used hei;j ; ei;ji = 1.) Furthermore, from (3.44) it followsthat ri miPj=1 �2i;j = 1: Since1 = ri miXj=1 �2i;j = 0@miXj=1 �i;j1A2 � mi miXj=1 �2i;j ;then ri � mi with equality if and only if all �i;j ; j = 1; :::;mi; are equal and, hence,equal to 1ri :We shall call an FDNDF-polynomial space X polynomial (cf. [82], [46]) if for anyi; i = 0; 1; ::; s(X); all �i;j , j = 1; :::;mi; are equal. Thus, for polynomial spaces theconstants ri in (2.39) are integers and equal to the dimensions of the spaces Vi; and(3.43) takes the form Qi(�(d(x; y))) = 1ri riXj=1 ei;j(x)ei;j(y): (3.53)Furthermore by (3.52), riXj=1 jei;j(x)j2 = ri for any x 2 X: (3.54)Fortunately, some metric spaces of signi�cant interest in coding theory, in particular,the Hamming space, the Johnson space and the unit Euclidean sphere, turn outto be polynomial. For the Hamming space the representation (3.6) gives a direct36

proof of orthogonality of the polynomials (2.44) and the property of the space to bepolynomial. However, in the general case a proof of this property is not a simpleproblem. The usual way to prove that a metric space X = fX; d(x; y)g is polynomialis based on the representation of the isometry group of the space by shift operatorsand the proof that zonal spherical functions of the representation are polynomials insome (substitution) function of distance d(x; y). We return to this problem at theend of the subsection.Consider an arbitrary code C in a polynomial space X with the standard substitu-tion �(d): Let �(C) = fd0 = 0; d1; :::; dsg; where s = s(C) is the number of distancesbetween distinct points of C and 0 < d1 < � � � < ds (hence, d(C) = d1). Then theminimal polynomial fC(t) can be represented as followsfC(t) = sYi=1 t� �(di)1� �(di) : (3.55)By Corollary 2.14 and (3.53) a weighted set C = (C;m) (Px2Cm(x) = jCj) in apolynomial space is a weighted � -design if and only if for any i = 1; :::; �;jCjB0i(C) = ri Xx;y2CQi(�(d(x; y)))m(x)m(y) = riXj=1 jXx2C ei;j(x)m(x)j2 = 0 (3.56)and hence if and only ifXx2C ei;j(x)m(x) = 0 for any i and j; i = 1; :::; �; j = 1; :::; ri: (3.57)Therefore for polynomial spaces X the de�nition (2.12) may be formulated (see also(2.56)) in the following form: a weighted set C = (C;m) is a weighted � -design if forany u(x) 2 S�i=0 Vi; ZX u(x)d� (x) = 1jCj Xx2C u(x)m (x) (3.58)or, equivalently, if for any polynomial f (t) in a real t of degree at most � and for anyy; z 2 X , ZX f(�(d(x; y)))d� (x) = 1jCj Xx2C f(�(d(x; z)))m (x) :(We used that all ei;j(x); i � 1; are orthogonal to e0;1(x) � 1 with respect to (3.9).)There exists one more equivalent de�nition in term of orthogonality of the functionsei;j(x) with weight m(x) on the (�nite) set C:Theorem 3.16. A weighted set C = (C;m) is a weighted � -design in a polynomialspace X if and only if for any integers i; j; k; l; 0 � i+ k � �; 1 � j � ri; 1 � l � rk;1jCj Xx2C ei;j(x)ek;l(x)m (x) = �i;k�j;l: (3.59)37

Proof. We use (3.54), (3.53), the Krein condition (3.47) and the equality riq0i;k = �i;kto prove the following equalities:1rirk riXj=1 rkXl=1 jXx2C ei;j(x)ek;l(x)m (x)� jCj�i;k�j;lj2= 1rirk riXj=1 rkXl=1 jXx2C ei;j(x)ek;l(x)m (x) j2 � jCj2ri �i;k= Xx;y2C �Qi(�(d(x; y)))Qk(�(d(x; y))) � q0i;k�m (x)m(y)= i+kXh=1 qhi;k Xx;y2CQh(�(d(x; y)))m (x)m(y) = jCj i+kXh=1 qhi;krh B0h(C):This completes the proof by (3.56).Now we extend some results on codes in Q-polynomial association schemes ofDelsarte [30] to the case of codes in polynomial spaces.Theorem 3.17. If for a code C in a polynomial space X; fC(t) = s(C)Pi=0 fiQi(t) and�(C) � s(C); then fj jCj = rj ; j = 0; 1; :::; �(C)� s(C):If fC(t) can be expanded over the system Q with positive coe�cients and for some � ,s(C) � � � s(X); fj jCj = rj ; j = 0; 1; :::; � � s(C);then �(C) � �:Proof. To prove the theorem it is su�cient to use the polynomials f(t) = fC(t)Qj(t)in (2.64), note that rj 1R�1 fC(t)Qj(t)d�(t) = fj ; and consider conditions for which theequalities rj Xx;y2C fC(�Q(d(x; y)))Qj(�Q(d(x; y))) = fj jCj2are valid.Theorem 3.18 (Absolute bound of Delsarte). For any code C in a polynomialspace X; jCj � s(C)Xi=0 ri (3.60)with equality if and only if fC(t) = Q1;0s (t) and C is a tight 2s-design where s = s(C):38

Proof. Let fC(t) = sPi=0 fiQi(t): For any y 2 C consider the continuous functionuy(x) = fC(�(d(x; y))) = sXi=0 firi riXj=1 ei;j(y)ei;j(x) (3.61)and note that uy(x) = � 1 if x = y;0 if x 6= y; x 2 C: (3.62)Since all these jCj functions belong to sSi=0Vi and are linearly independent, we have(3.60). Equality holds if and only if for any i = 0; 1; :::; s; j = 1; :::; ri and any y 2 Cthere exist constants ci;j(y) such thatei;j(x) = Xy2C ci;j(y)uy(x):By (3.62), ci;j(y) = ei;j(y); and by (3.61) the last equalities are equivalent to the factthat firi Xy2C ei;j(y)ek;l(y) = �i;k�j;l:This completes the proof by taking into account (3.54) and Theorem 3.16.The idea of the proof of Theorem 3.18 was extended to prove many similar results(see the review paper [18] and [7]).Theorem 3.19. Any code C in a polynomial space X such that �(C) � s(C)� 1 isdistance invariant.Proof. For any x 2 X and any polynomial f(t) = hPi=0 fiQi(t) of degree h � �(C) by(2.8), (3.53), and (3.57) we haves(C)Xi=0 Bdi(x;C)f(�(di)) = Xy2C f(�(d(x; y)))= hXi=0 firi riXl=1 ei;l(x)Xy2C ei;l(y) = f0jCj: (3.63)In particular, for any j = 1; :::; s; s = s(C); we consider the Lagrange polynomialf (j)C (t) = fC(t)(t� �(dj))f 0C(�(dj)) (3.64)39

of degree h = s � 1 (here f 0C(t) is the derivative of fC(t)) satisfying the followingproperty: f (j)C (�(di)) = �i;j ; i = 1; :::; s;and assume that f (j)C (t) = s�1Xi=0 fj;iQi(t): (3.65)Using (3.63) and the fact that B0(x;C) = 1 for all x 2 C; we obtainBdj (x;C) = fj;0jCj � f (j)C (1); j = 1; :::; s; (3.66)and hence Bdj (x;C) does not depend on x 2 C:Corollary 3.20. Any code C in a �nite polynomial space X such that �(C) = s(X)coincides with the whole space X .Proof. If we use the polynomial fX(t) with s = s(X) (see (3.55)) in (3.63) whereh = s(X) = �(C) and also (2.42), (2.35), we obtain that for any x 2 XB0(x;C) = jCj s(X)Xi=0 fX(�(di))wi = jCjw0 = jCjjX jand hence C = X .Theorem 3.21. Any code C in a polynomial space X with the standard substi-tution function�(d) such that �(C) � 2s(C) � 2 forms a distance-regular poly-nomial space with the same substitution function. The intersection numbers pki;j ;i; j; k 2 f0; 1; :::; s(C)g (see (2.9)) can be de�ned by means of (3.64) and (3.65) asfollows: pki;0 = pk0;i = �i;k; p0i;j = �i;j �fj;0jCj � f (j)C (1)� if j � 1; (3.67)and for i; j; k 2 f1; :::; s(C)g ;pki;j = jCj s�1Xl=0 fi;lfj;lrl Ql(�(dk))� f (i)C (1)�j;k � f (j)C (1)�i;k: (3.68)Proof. By Theorem 3.19 Bdj (x;C) does not depend on x 2 C (see (3.66)) and wedenote it by Bdj (C); j = 0; 1; :::; s; s = s(C): To prove that C is a polynomial spacewe shall construct for any i = 0; 1; :::; s polynomials Qi(t) of degree i; Qi(1) = 1; andpositive constants ri such thatrijCj sXj=0Qi(�(dj))Qk(�(dj))Bdj (C) = �i;k; (3.69)40

and also construct an orthonormal basis of functions vi;j(x) on C; i = 0; 1; :::; s; j =1; :::; ri; with respect to the inner producthu;vi = 1jCj Xx2C u(x)v(x) (3.70)such that Qi(�(d(x; y))) = 1ri riXj=1 vi;j(x)vi;j(y): (3.71)For any i = 0; 1; :::; s� 1 we putri = ri; Qi(t) = Qi(t); vi;j(x) = ei;j(x); j = 0; 1; :::; ri;and let rs = jCj� Ps�1i=0 ri andrsQs(t) = jCj sYj=1 t� �(dj)1� �(dj) � s�1Xi=0 ri Qi(t): (3.72)By Theorem 3.16 vi;j(x); i = 0; 1; :::; s � 1; j = 0; 1; :::; ri; are orthonormal withrespect to (3.70) and hence rs � 0: In fact rs > 0 since otherwise by Theorem 2.16C is a tight (2s � 2)-design which by Lemma 2.18 cannot have s distinct nonzerodistances. Using the orthogonalization process we can de�ne the remaining functionsvs;j(x); j = 0; 1; :::; rs; so that all jCj functions vi;j(x); i = 0; 1; :::; s; j = 0; 1; :::; ri;have the property 1jCj Xx2C vi;j(x)vk;l(x) = �i;k�j;l: (3.73)By the construction, (3.71) holds for i = 0; 1; :::; s�1: Since 1jCj sPi=0Prij=1 vi;j(x)vi;j (y)is the identity matrix of order jCj and, hence, coincides with sQj=1 �(d(x;y))��(dj)1��(dj) ; (3.71)is also true for i = s according to (3.72). The orthogonality conditions (3.69) followfrom (3.71), (3.73) and the fact that C is distance invariant. Thus C forms a poly-nomial space with the substitution function �(d) which is standard if and only if Cis diametrical. Now we prove that for any x; y 2 C and any i; j 2 f0; 1; :::; s(C)gthe intersection numbers px;yi;j (see (2.9)) depend only on d(x; y) and are de�ned bymeans of (3.67) and (3.68). By (3.66) the only case we need to consider is wheni; j 2 f1; :::; s(C)g and x 6= y: In this casepx;yi;j = Xz2Cnfx;yg f (i)C (�(d(x; z)))f (j)C (�(d(y; z))):On the other hand, if d(x; y) = dk, then by (3.53), (3.65) and Theorem 3.16,Xz2C f (i)C (�(d(x; z)))f (j)C (�(d(y; z))) =Xz2C s�1Xl=0 s�1Xm=0 fi;lfj;mQl(�(d(x; z)))Qm(�(d(z; y)))41

= s�1Xl=0 s�1Xm=0 fi;lfj;mrlrm riXa=1 rmXb=1 el;a(x)em;b(y)Xz2C em;b(z)el;a(z)= jCj s�1Xl=0 s�1Xm=0 fi;lfj;mrlrm riXa=1 rmXb=1 el;a(x)em;b(y)�l;m�a;b= jCj s�1Xl=0 fi;lfj;lrl Ql(�(dk)):This implies (3.68) and completes the proof.The proof of Theorem 3.21 in fact uses the following equivalent de�nition of apolynomial space which are useful in proving spaces are polynomial. Suppose for acompact metric spaceX = (X; d(x; y); �) with a standard substitution �(d) there exist�nite-dimensional subspaces Vi; i = 0; 1; :::; s(X); of continuous functions which arepairwise orthogonal with respect to inner product (3.9) and V0 consists of constants.Then X is polynomial (with respect to �(d)) if there exist polynomials Qi(t) of degreei (i = 0; 1; :::; s(X)) such that for any x; y 2 XQi(�(d(x; y))) = 1ri riXj=1 ei;j(x)ei;j(y): (3.74)where ri = dimVi and fei;j(x); j = 1; :::; rig is an orthonormal basis of Vi (the right-hand side of (3.74) does not depend on a choice of the basis). It is easy to verify thatthe system of polynomials fQi(t); i = 0; 1; :::; s(X)g satis�es the orthogonality andnormalization conditions (2.39)-(2.40) and coincides with the system Q constructedin Section 2.2.Consider for a �nite or in�nite connected compact metric space X = (X; d(x; y))an isometry group G, i.e., a group of continuous one-to-one mappings X onto Xsuch that d(gx; gy) = d(x; y) for any x; y 2 X and g 2 G. A space X is calleddistance-transitive (with respect to G) if d(x1; y1) = d(x2; y2) implies the existence ofan isometry g 2 G with gx1 = x2 and gy1 = y2. In particular (the case x1 = y1 andx2 = y2), this means that G acts transitively on X , i.e., for any x; y 2 X there existsg 2 G such that gx = y. Therefore, on a distance-transitive space X there exists aunique normalized invariant measure, the Haar measure, � (�(gA) = �(A) for anymeasurable A � X and any g 2 G; �(X) = 1) and X with this measure is distanceinvariant. In the case of a distance-transitive space X we consider the quasi-regulargroup G representation L(g) in the Hilbert space L2(X;�) by translations, de�ned asfollows L(g)u(x) = u(g�1x): (3.75)The representation is decomposable (see, for example, [120]) into an orthogonal directsum of pairwise non-equivalent irreducible representations Li(g) acting on invariant�nite dimensional subspaces Vi of continuous functions (including the space V0 ofconstant functions). Since the subspaces Vi are invariant (with respect to the action42

of G) and X is distance-transitive, there exist real (so-called zonal spherical) functions�i(d); i = 0; 1; ::: , such that for any x; y 2 X�i(d(x; y)) = 1ri riXj=1 ei;j(x)ei;j(y); (3.76)where ri = dimVi and fei;j(x); j = 1; :::; rig is an arbitrary orthonormal basis ofVi: If there exist an ordering of the spaces Vi; polynomials Qi(t) of degree i; anda (standard) substitution function �(d) such that �i(d) = Qi(�(d)) for all i andd 2 �(X); then X is polynomial with respect to �(d) and fQi(t); i = 0; 1; :::; s(X)gcoincides with the system Q constructed in Section 2.2. More detailed description ofthis approach is contained in [30], [57], [108], [111], [10].This approach was used to prove that some families of distance-transitive graphsare polynomial [30]-[32], [109], [110]. However, distance-transitive polynomial graphshave not yet been classi�ed (the books [8], [23] contain detailed information about theproblem). All in�nite connected compact metric spaces which are distance-transitivewith respect to their full isometry group (also called \two-point homogeneous") wereclassi�ed by Wang [121]. Namely, they are the unit sphere Sn�1 in Rn and thereal RPn�1, the complex CPn�1, the quaternionic HPn�1 projective spaces (seeExamples 2.3, 2.10, 2.4, 2.11) and Cayley's plane OP 2. In each of these cases thereis known [26], [95], [44], [42], [3], [59], [53] to exist an ordering of the spaces Vi andparameters � and � such that�i(d) = Qi(�(d)) = P�;�i (�(d)); i = 0; 1; ::: (3.77)where P�;�i (t) are Jacobi polynomials of degree i and �(d) is a continuous decreasingfunction with property (2.32). So they all are polynomial. The parameters � and �,and the standard substitution �(d) for Sn�1 and the projective spaces are given inthe examples mentioned above.3.3. A �-packing and �-design problem for systems of orthogonal polyno-mialsTheorems 3.14 and 3.15 show that the problem of obtaining universal bounds for codesand designs in polynomial spaces, in the framework of the method considered, reduceto the following extremum problems for the corresponding systems Q of orthogonalpolynomials satisfying the Krein condition.The �-packing problem: for any �; �1 � � < 1; �nd the in�mum of Q(f) =f(1)f0(Q) over the class of polynomials f(t) = s(X)Pi=0 fi(Q)Qi(t) such that f0(Q) > 0;fi(Q) � 0 for i = 1; :::; s(X) andf(t) � 0 for � 1 � t � �: (3.78)43

The � -design problem: for any integer �; � = 1; :::; s(X); �nd the supremumof Q(f) = f(1)f0(Q) over the class of polynomials f(t) = s(X)Pi=0 fi(Q)Qi(t) such thatf0(Q) > 0; fi(Q) � 0 for i = � + 1; :::; s(X) andf(t) � 0 for � 1 � t � 1: (3.79)In Section 2.3 we described the solution of the restricted (in the class of polynomi-als of degree at most �) � -design problem for any orthogonal systems Q. This solutiongives rise to the universal bound (2.75) for (weighted) � -designs in arbitrary compactmetric spaces. Now we give a preliminary description of a solution of the restricted(in the class of polynomials of degree at most h(�) where the function h(�) will bede�ned below) �-packing problem for orthogonal systems Q satisfying the Krein con-dition. This solution gives rise to universal bounds for codes in polynomial spaces.We also verify that the solution of the restricted � -design problem can be obtainedfrom the optimal polynomials for the restricted �-packing problem for special valuesof �.We consider a �nite or countable system Q = fQi(t); i = 0; 1; :::; sg of orthogonalpolynomials on [�1; 1] with the normalization and orthogonality conditions (2.40),(2.42) or (2.43). We use some properties of adjacent systems Qa;b of orthogonalpolynomials de�ned by (2.67) or (2.68), of the largest roots ta;bi of polynomials Qa;bi (t);i = 0; 1; :::; sa;b; and of the function (kernel) T a;bi (x; y) de�ned by (2.69) which will beproved in Section 5.For any k; 1 � k < s; the following inequalities holdt1;1k�1 < t1;0k < t1;1k where t1;10 = �1: (3.80)This means that the half-open interval [�1; 1) in the case of a countable system Qor half-open interval4 [�1; t1;1s�1) in the case of a �nite Q is partitioned into half-open intervals [t1;1k�1; t1;0k ) and [t1;0k ; t1;1k ); k = 1; :::; s� 1: Enumerate in succession allthese half-open intervals from the left to the right by positive integers. For any �,�1 � � < t1;1s�1 (we put t1;1s�1 = 1 if s =1), denote by h(�) the number of the (unique)half-open interval containing �. Let k(�) = k when � 2 [t1;1k�1; t1;0k ) or � 2 [t1;0k ; t1;1k );and let "(�) = 0 if � 2 [t1;1k�1; t1;0k ) for some k and "(�) = 1 if � 2 [t1;0k ; t1;1k ) for somek. Then it is clear that h(�) = 2k(�)� 1 + "(�): (3.81)For any �, �1 � � < t1;1s�1; consider the polynomialf (�)(t) = (t� �)(t+ 1)" �T 1;"k�1(t; �)�2 (3.82)4For a �nite metric space X with the standard substitution �(d) and �(X) = f0; d1; :::; dsg where0 < d1 < � � � < ds and s = s(X), we have A(X; d1) = jXj and t1;1s�1 = �(d1). Since �(d) decreaseswith d it is su�cient to consider the �-packing problem for the system Q for �1 � � < t1;1s�1.44

where " = "(�) and k = k(�): Note that the polynomial (3.82) has degree h(�) andsatis�es property (3.78). We shall prove that f (�)(1) > 0 and f (�)0 (Q) > 0 and thefunction LQ(�) = (f (�)) = f (�)(1)f (�)0 (Q)can be represented in the formLQ(�) = �1� 1Q1(�1)�" 1� Q1;"k�1(�)Q0;"k (�) ! k�1Xi=0 r0;"i : (3.83)The function LQ(�) grows with � and takes the following values at the left ends ofthese half-open intervalsLQ(t1;�l��) = �1� 1Q1(�1)�� l��Xi=0 r0;�i (3.84)where l is an integer and � 2 f0; 1g: In particular,LQ(�1) = 1� 1Q1(�1) : (3.85)For � = t1;0k we have k(�) = k, "(�) = 1 and the polynomial f (�)(t) has factor t+ 1.For � = t1;1k�1 we have k(�) = k, "(�) = 0; however, we shall see that the polynomialf (�)(t) is also divisible by t+1. In the both cases the polynomial f(t) = f (�)(t)=(t+1)satis�es property (3.78) and (f) = (f (�)) = LQ(�):We shall see that the polynomials f (�)(t) have the following extremum property:for any �, �1 � � < t1;1s�1; and any polynomial f(t) of degree at most h(�) such thatf0(Q) > 0 and (3.78) holds, (f) = f(1)f0(Q) � LQ(�) (3.86)with equality if and only if f(t) is proportional to f (�)(t) or f (�)(t)=(t + 1) in thecases when � = t1;�l for some integer l and � 2 f0; 1g:In order to prove that f (�)(t) is the optimal solution of the �-packing problem inthe class of polynomials of degree at most h(�) we should show that all coe�cientsof f (�)(t) are nonnegative. Analogously to (2.54) for any polynomial f(t) we de�necoe�cients fi(Qa;b) = ra;bi 1Z�1 f(t)Qa;bi (t)d�a;b(t); i = 0; 1; :::; sa;b: (3.87)Denote by F�(Qa;b) (respectively, F>(Qa;b)) the set of polynomials f(t) 2 F [t] suchthat fi(Qa;b) � 0; i = 0; 1; :::; sa;b45

(respectively, such thatfi(Qa;b) > 0; i = 0; 1; :::; h; and fi(Qa;b) = 0; i = h+ 1; :::; sa;bfor some h; 0 � h � sa;b). Then the fact that the system Q = fQi(t); i = 0; 1; :::; sgsatis�es the Krein condition can be written as followsQi(t)Qj(t) 2 F�(Q) for any i; j; 0 � i; j � s: (3.88)Note that t+ 1 2 F>(Q) (3.89)since Q0(t) � 1 and t+ 1 = 2Q1(t)�Q1(�1)1�Q1(�1) ; and thatf(t)g(t) 2 F>(Q) if f(t) 2 F>(Q) and g(t) 2 F>(Q): (3.90)Moreover, from (2.69)-(2.70) and the Christo�el-Darboux formula (see Theorem 5.4)it follows that Q1;0i (t) 2 F>(Q); (t+ 1)Q0;1i (t) 2 F�(Q): (3.91)We say that the system Q = fQi(t); i = 0; 1; :::; sg satis�es the strengthened Kreincondition if together with (3.88) the following holds(t+ 1)Q1;1i (t)Q1;1j (t) 2 F>(Q) for any i; j; 0 � i; j � s1;1: (3.92)Lemma 3.22. A system Q = fQi(t); i = 0; 1; :::g for any compact metric spacesatis�es the strengthened Krein condition if for some � and � such that � � � � � 12 ;Qi(t) = P�;�i (t); i = 0; 1; ::: ,where P�;�i (t) are Jacobi polynomials normalized by P�;�i (1) = 1 (see (2.48)).Proof. By the Gasper Theorem [43] for � � �; �+ � + 1 � 0; the system fP�;�i (t);i = 0; 1; :::g satis�es the Krein condition. Since Q1;1i (t) = P�+1;�+1i (t); it is true forthe system fQ1;1i (t); i = 0; 1; :::g:We shall see in Section 5 (Corollary 5.25) that thereexist positive constants li and mi such thatQ1;1i (t) = liQ0;1i (t) +miQ1;0i (t): (3.93)This completes the proof using (3.89)-(3.91).Lemma 3.22 shows that the systems Q for in�nite distance-transitive polynomialspaces (in particular, for the Euclidean sphere) satisfy the strengthened Krein condi-tion.A �nite polynomial space X = (X; d(x; y)) with a standard substitution �(d) iscalled decomposable [68] if for some h there exist metric subspaces Xi = (Xi; d(x; y));i = 1; :::; h; of X such that: 46

(i) X = hSi=1Xi;(ii) all subspaces Xi are isometric to a single metric space eX = ( eX; d(x; y)) whichis polynomial with the same �(d),(iii) for any x; y 2 X the number of subspaces Xi containing both x and y is equalto �(d(x; y)) + 12 h j eXjjX j : (3.94)Let X be a decomposable space and X1; :::; Xh be the subspaces mentioned in thede�nition. For any C � X we say that C \Xj is the projection of C onto Xj and, inparticular, Xj is the projection of X (onto Xj). Notice that the space eX (and anyXj) is polynomial with respect to �(d), which is not standard for eX , since from (2.32)and (3.94) it follows that D( eX) = D(X)� 1:(We shall not standardize the substitution function in eX in order to have a simpleformulation of Theorem 3.24.) The parameters of the space eX, which are analogousto vi; ri; and Qi(t); i = 0; 1; :::; D; of the space X , are denoted by evi; eri; and eQi(t);i = 0; 1; :::; D� 1; respectively.Example 3.23. For the Hamming spaceHnv with standard substitution �(d) = 1� 2dnconsider h = nv metric subspaces obtained from Hnv by �xing any value 0; 1; :::; v� 1in any of the n coordinates. Each of these subspaces is isometric to Hn�1v and henceis polynomial with the same �(d): Any points x; y 2 Hnv such that d(x; y) = i belongto �(i) + 12 nv jHn�1v jjHnv j = n� isubspaces. Hence, the Hamming space Hnv is decomposable. For the Johnson spaceJnw with standard substitution �(d) = 1 � 2dw consider h = n metric subspaces ofJnw consisting of w-sets which contain a �xed element of f1; :::; ng. Each of thesesubspaces is isometric to Jn�1w�1 and hence is polynomial with the same �(d): Anypoints x; y 2 Jnw such that d(x; y) = i belong to�(i) + 12 n jJn�1w�1jjJnwj = w � isubspaces. Hence, the Johnson space Jnv is also decomposable.Theorem 3.24 ([70]). Let X be a decomposable polynomial space of diameter D.Then jX j = LQ(�1)j eXj and for i = 0; 1; :::; D� 1;evi = �(i) + 12 vi;47

eri = r0;1i ; eQi(t) = Q0;1i (t):Thus for a decomposable polynomial space the system Q0;1 satis�es the Kreincondition as well. Since Q1;1i (t) 2 F>(Q0;1) (see, for example, (2.73) for � = 1; i =l � 1), this gives rise to the following statement [70].Lemma 3.25. The system Q for any decomposable polynomial space satis�es thestrengthened Krein condition.Thus the systemsQ for polynomial spaces which are in�nite and distance-transitiveor �nite and decomposable satisfy the strengthened Krein condition. In Section 5.4we prove that if for a polynomial space the system Q satis�es the strengthened Kreincondition, then for any �, �1 � � < t1;1s�1; all coe�cients of the polynomial f (�)(t) ofdegree h(�) = 2k(�)� 1 + "(�) over Q are positive (i.e., f (�)(t) 2 F>(Q)).Theorem 3.14 and the results mentioned above allow us to obtain the followingmain bound for any code C in a polynomial space X with a standard substitution�(d) and with the system Q de�ned by Theorem 2.7 which satis�es the strengthenedKrein condition: jCj � LQ(�(d(C))): (3.95)We shall also �nd necessary and su�cient conditions for equality in (3.95) in termsof the values �(C) and fC(t).The author assumed that f (�)(t) 2 F>(Q) for any system Q corresponding to apolynomial metric space (and, hence, (3.95) is true without the restriction on thestrengthened Krein condition) but could not prove this assumption. In the generalcase it is not known whether f (�)(t) 2 F>(Q) when � lies inside the half-open intervalwith even number h(�) = 2k(�) � 4 (and hence "(�) = 1). In this connection for any�, �1 � � < t1;0s�1; we also consider the polynomialsef (�)(t) = (t� �)�T 1;0k�1(t; �)�2 (3.96)of odd degree 2k � 1; where k is uniquely determined by t1;0k�1 � � < t1;0k ; and thefunction eLQ(�) = ( ef (�)) = 1� Q1;0k�1(�)Qk(�) ! k�1Xi=0 ri: (3.97)It can be shown that ef (�)(t) 2 F>(Q) for any system Q satisfying the Krein condition,and that eLQ(�) is a continuous (and even di�erentiable [67]) function of �: This givesthat for any code C in a polynomial space,jCj � 1� Q1;0k�1(�)Qk(�) ! k�1Xi=0 ri: (3.98)where � = �(d(C)) and t1;0k�1 � � < t1;0k : 48

The polynomialsh(�)(t) = (t� �)�T 0;0k�1(t; �)�2 where t0;0k�1 < � < t0;0k (3.99)of odd degree 2k � 1 were used before in [79] for the Hamming and Johnson spacesand in [57] for the Euclidean sphere and other distance-transitive in�nite compactspaces. They have the property h(�)(t) 2 F>(Q) and give rise to the universal boundjCj � � 1Qk�1(�) � 1Qk(�)� k�1Xi=0 riQi(�); (3.100)where � = �(d(C)): For � > t1;01 this bound is worse compared to (3.95) and (3.98)and cannot be attained (for �1 � � � t1;01 these bounds are obtained by using thepolynomial t� � and coincide). However, in a certain asymptotic process these threebounds give rise to the same asymptotic result.Note the signi�cant special cases of the universal bounds (3.95) and (3.98). If� = t1;�l�� then k(�) = l; "(�) = 1 � �; and h(�) = 2l � �: Using (3.84), the notationda;bi = ��1(ta;bi ) (see (2.71)), and monotonicity of the functions (3.83) and (3.97), wehave that for any code C in a polynomial space,jCj � LQ(t1;�l��) = �1� 1Q1(�1)�� l��Xi=0 r0;�i if d(C) � d1;�l��: (3.101)Consider now the polynomialsf(t) = t� �t+ 1 f (�)(t) where � = t1;�l�� (3.102)of degree h(�) = 2l � � which by the construction satisfy (3.79). Moreover, we shallsee that the polynomials (3.102) up to a constant multiple are equal to polynomialsg(2l��)(t) de�ned by (2.72) and have the property(f) = (f (�)) = LQ(t1;�l��):Thus the polynomial (3.102) is a solution of the restricted (2l��)-design problem (cf.Theorem 2.16). We shall verify in Section 5 that the optimality of (3.102) and theoptimality of f (�)(t) for the restricted (2l��)-design and �-packing problems, respec-tively, are consequences of the same result in the theory of orthogonal polynomials.For a polynomial metric space X we de�ne one more function (d) of an integerd as follows: (d) = t1;�l�� if d = 2l+ 1� � (3.103)where l is an integer and � 2 f0; 1g . Since t1;�l�� is the left end of the half-open intervalnumbered by d� 1 = 2l � �; from (3.80) it follows that (d) increases with d. Usingthe function (d) we can express the bound (2.75) of Theorem 2.16 as followsjCj � LQ( (d)) if d0(C) � d: (3.104)49

Furthermore, we can rewrite (3.101) in the formjCj � LQ( (d)) if �(d(C)) � (d):and formulate the following known (see, for example [82]) conjecture.Conjecture 3.26. For any code C in a polynomial space X;LQ( (d0(C))) � jCj � LQ( (2s(C)� �(C) + 1)) (3.105)with equality in either of the bounds if and only ifd0(C) = 2s(C)� �(C) + 1and fC(t) = � t+ 12 ��(C)Q1;�(C)s(C)��(C)(t):This statement is true with respect to the lower bound (see Theorem 2.16 andLemma 2.18) and with respect to the upper bound when �(C) = 0 (see Theorem3.18). Thus one has only to prove the upper bound in (3.105) for diametrical codes(�(C) = 1) and the fact that the stated conditions of its attainability are necessaryin this case (the su�ciency of the conditions follows from Lemma 2.18).Theorem 3.27. Conjecture 3.26 is true for all decomposable polynomial spaces.Proof. As was noted before, we need to prove only the upper bound (3.105) fordiametrical codes and the necessary conditions of its attainability. Let C be a dia-metrical code in a decomposable space X and let s(C) = s. Since the function �(d)is standard, fC(t) = 1+t2 g(t), where g(t) is a polynomial of degree s � 1 such thatg(1) = 1: From (3.94) and Theorem 3.24 it follows that every point x 2 X belongs toh=LQ(�1) projections among h ones X1; :::; Xh of the space X: HencehjCj = LQ(�1) hXj=1 jCj j; (3.106)where Cj = C\Xj is the projection of C ontoXj ; j = 1; :::; h. Since s(Cj) � s(C)�1,using Theorem 3.24 and the absolute bound (3.60) for the space Xj ; we havejCj j � s�1Xi=0 r0;1i : (3.107)Moreover, if the bound (3.107) is attained then g(t) = eQ1;0s�1(t) and hence (see (2.70))g(t) = Q1;1s�1(t). From (3.106) and (3.107) it follows thatjCj � LQ(�1) s�1Xi=0 r0;1i (3.108)50

and if the bound (3.108) is attained, then for any projection Cj the bound (3.107)is attained and fC(t) = 1+t2 Q1;1s�1(t): Note that (3.108) coincides with the upperbound in (3.105) for the case considered. When the bound (3.108) is attained andfC(t) = 1+t2 Q1;1s�1(t); we consider the polynomial f(t) = 1+t2 �Q1;1s�1(t)�2. By Lemma3.25 all its coe�cients fi = ri s(X)Pj=0 f(�(dj))Qi(�(dj))wj ; i = 0; 1; :::; 2s�1; are positive.Since f(1) = 1 and f0 = jCj�1 (see (2.72) and (2.75)), from (2.64) and (2.60) it followsthat 2s�1Xi=1 firiB0i(C) = 0:This means that d0(C) � 2s(C) and hence d0(C) = 2s(C) by Lemma 2.17.This conjecture was also proved for in�nite distance-transitive compact metricspaces [35], [34], [54], [53] (in particular, for the Euclidean sphere). The question ofwhether this conjecture is true for all polynomial spaces is open now (1997). It seemsbelievable because Lemma 2.18 would be a consequence of the statement.4. Duality in bounding optimal sizes of codes and designs inpolynomial graphs4.1. Polynomial graphsIn this section we study codes and designs in �nite polynomial metric spaces whichare P - and Q-polynomial association schemes [30], [57], [8], [23]. As we can see later itis convenient to consider such association schemes as graphs with the path metric andcall them polynomial graphs. In this connection we use a de�nition for associationschemes which is close to the de�nition of metric spaces.A (symmetric) association scheme (with D classes) fX; d(x; y)g is a �nite set Xwith a given function d(x; y) which is de�ned for any x; y 2 X , takes values 0; 1; : : : ; Dand has the following properties :1. d(x; y) = 0 if and only if x = y ;2. d(x; y) = d(y; x) for any x; y 2 X ;3. for any x; y 2 X and any i; j 2 f0; 1; : : : ; Dg, the number of points z suchthat d(x; z) = i; d(z; y) = j depends on d(x; y) only (this number is called anintersection number and denoted by pki;j , where k = d(x; y)).Note that any distance-regular code C; in a (�nite or in�nite) metric space Z witha metric dZ(x; y) such that �(C) = fd0 = 0; d1; :::; dsg ; is an association scheme withD = s classes if we put d(x; y) = i when dZ(x; y) = di. In particular, by Theorem3.21 all codes C in a polynomial space such that �(C) � 2s(C) � 2 are associationschemes and their intersection numbers pki;j can be calculated by (3.67) and (3.68).51

For association schemes, pki;j = pkj;i and p0i;i is the number of points y 2 X suchthat d(x; y) = i for a �xed element x 2 X . The numbers p0i;i are denoted by vi andcalled valencies. In general the function d(x; y) does not satisfy the triangle inequalityd(x; y) � d(x; z) + d(z; y). However, for many signi�cant examples of associationschemes the function d(x; y) has this property and is a metric. In particular, theHamming space Hnv (see Example 2.1), with the metric d(x; y) being equal to thenumber of places where x and y di�er, forms an association scheme with D = nclasses. As another example of an association scheme with D = w consider theJohnson space Jnw, w � n=2 (see Example 2.2) consisting of �nw� w-subsets of an n-setwith the metric d(x; y) = w � jx \ yj.Using the adjacency matrices Ai; i = 0; 1; : : : ; D, of order jX j de�ned by(Ai)x;y = � 1 if d(x; y) = i,0 otherwise, (4.1)the de�nition of an association scheme can be expressed byA0 = I; DXi=0 Ai = J; Ai = ATi ; AiAj = DXk=0 pki;jAk; (4.2)where I is the identity matrix, J is the matrix with entries all equal to one and ATis the transpose of A.The matrices Ai are linearly independent and generate a (D+1)-dimensional (overR ) commutative algebra A of symmetric matrices, which is called the Bose-Mesneralgebra. We consider the jX j-dimensional vector space V = fu(x) : X ! Rg of realfunctions on X with the inner producthu;vi = 1jX j Xx2X u(x)v(x):It is known [30], [8], [23] that for an association scheme with D classes (as forthe (D + 1)-dimensional commutative algebra of symmetric matrices) there exists adecomposition V = V0 + : : :+ VDof V into a direct sum of pairwise orthogonal subspaces Vi, i = 0; 1; : : : ; D, whereeach Vi is a maximal common eigenspace of A0; A1; : : : ; AD . We can assume thatV0 consists of constants only since the eigenvector of all ones can belong only to aone-dimensional common eigenspace.Let ri = dimVi; i = 0; 1; : : : ; D; r0 = 1, and fvi;j(x); j = 1; : : : ; rig be anyorthonormal basis of Vi. The matricesEi(x; y) = 1jX j riXj=1 vi;j(x)vi;j(y); i = 0; 1; : : : ; D; (4.3)52

do not depend on the choice of the bases of Vi, possess the propertiesEiEj = Ei�i;j ; E0 = 1jX jJ; DXi=0 Ei = I; (4.4)and hence form the basis of irreducible idempotents of A. It follows that there existtwo non-degenerate matrices P = (Pi;j) and Q = (Qi;j) of order D + 1 such thatAj = DXi=0 Pi;jEi; j = 0; 1; : : : ; D; (4.5)Ej = 1jX j DXi=0 Qi;jAi; j = 0; 1; : : : ; D: (4.6)The equations (4.3)-(4.5) show that the column space of Ei is an eigenspace of eachAj , and the corresponding eigenvalue Pi;j has the multiplicity ri = rank Ei = tr Ei.In particular, by (4.1), (4.4)-(4.5) P0;i is equal to the valency vi = p0i;i, and by (4.3)and (4.6) Q0;i = ri. From (4.5) and (4.6) it follows thatPQ = QP = 1jX jI (4.7)where I here is the identity matrix of order D + 1. Considering tr(AjEi) and usingtr(AiAj) = vijX j�i;j ; by (4.1) and (4.2), we also get thatPi;jri = Qj;ivj : (4.8)It is clear that an association scheme is a metric space with distance d(x; y) whenthe triangle inequality holds for this function. Delsarte proved [30] that it holds ifthere exists polynomials pj(�) of degree j; j = 1; : : : ; D, of a real variable � such thatAj = pj(A1) or, in other words (see (4.5)),Pi;j = pj(Pi;1); i = 0; 1; : : : ; D (4.9)Such an association scheme is called P -polynomial or metric.There is another description of metric association schemes in terms of graphs. Thevertex set X of any undirected graph � can be considered as a metric space withthe path metric d�(x; y) equal to the number of edges in the shortest path from xto y. An undirected connected graph � with the vertex set X is called distance-regular if for any x; y 2 X the number of vertices z such that d�(x; z) = 1; d�(y; z) =d�(x; y)�1 and the number of vertices z such that d�(x; z) = 1; d�(y; z) = d�(x; y)+1depend on d�(x; y) only5. Delsarte [30] proved that for any distance-regular graph5It is true that a distance-regular graph � is a distance-regular metric space with the pathmetric d�(x; y) and its intersection numbers pki;j are uniquely de�ned by pk1;k�1 and pk1;k+1;i; j; k 2 f0; 1; :::;Dg (see [30], [8] or [23]). 53

� with vertex set X; fX; d�(x; y)g is a metric association scheme, and for any metricassociation scheme fX; d(x; y)g; the graph � with the vertex set X and the adjacencymatrix A1 is a distance-regular graph, and d�(x; y) = d(x; y). Thus, there is one-to-one correspondence between metric association schemes with D classes and distance-regular graphs of diameter D.An association scheme fX; d(x; y)g with D classes is called Q-polynomial, or co-metric, if there exist polynomials qj(�) of degree j; j = 0; 1; : : : ; D, of a real variable� such that Qi;j = qj(Qi;1); i = 0; 1; : : : ; D: (4.10)Hereafter we consider P - and Q-polynomial association schemes (or Q-polynomialdistance-regular graphs [23]) which are referred to as polynomial graphs. Note thatQi;1 (and Pi;1) are di�erent for di�erent i; i = 0; 1; : : : ; D, since otherwise the matrices(4.6) (respectively (4.5)) would be linearly dependent. We introduce an additionalrestriction that Qi;1 and Pi;1 decrease with i, which is ful�lled for many (but not all)polynomial graphs. Let �Q(d) and �P (d) be continuous decreasing functions in a realvariable d (the substitutions) such that for any d = 0; 1; : : : ; D,�Q(d) = 1� 2 r1 �Qd;1r1 �QD;1 ; �P (d) = 1� 2 v1 � Pd;1v1 � PD;1By the construction and the assumption�Q(D) = �1 � �Q(d) � �Q(0) = 1; �P (D) = �1 � �P (d) � �P (0) = 1: (4.11)The following examples are well known [29]-[32].Example 4.1. For the Hamming space Hnv :vi = ri = �ni�(v � 1)i; i = 0; 1; : : : ; n;Qi;k = Pi;k = Kn;vk (i); i; k = 0; 1; : : : ; n;where Kn;vk (z) is the Krawtchouk polynomial of degree k, de�ned by (2.44) and, inparticular, Qi;1 = Pi;1 = (v � 1)n� vi:Thus Hnv is a polynomial graph with�Q(d) = �P (d) = 1� 2 dn:Example 4.2. For the Johnson space Jnw with 1 � w � n=2 :vi = �ni��n�wi �; i = 0; 1; : : : ; wri = �ni�� � ni�1�; i = 1; : : : ; w (r0 = 1);54

Qi;k = rk kXj=0(�1)j �kj��n+1�kj ��wj ��n�wj � �ij�;Pi;k = kXj=0(�1)k�j�w � jk � j��w � ij ��n� w + j � ij �= kXj=0(�1)j�ij��w � ik � j��n� w � ik � j �;and, in particular, Qi;1 = n�1� niw(n�w)� ;Pi;1 = w(n� w)� i(n+ 1� i):Thus Jnw is a polynomial graph with�Q(d) = 1� 2 dw ; �P (d) = 1� 2 d(n+ 1� d)w(n+ 1� w) :Now we introduce systems fQi(t)g and fPi(t)g of polynomials in a real t; 0 �t � 1, which are obtained from polynomials (4.10) and (4.9) by change of variablesas follows: rjQj(t) = qj( t(r1 �QD;1) + r1 +QD;12 );vjPj(t) = pj( t(v1 � PD;1) + v1 + PD;12 );and hence for any i; j = 0; 1; : : : ; D,rjQj(�Q(i)) = Qi;j ; vjPj(�P (i)) = Pi;j : (4.12)By (4.5)-(4.12) we have the following equalitiesri DXd=0Qi(�Q(d))Qj(�Q(d))vd = �i;j jX j; Qi(1) = 1; (4.13)vi DXd=0Pi(�P (d))Pj(�P (d))rd = �i;j jX j; Pi(1) = 1; (4.14)Qi(�Q(d)) = Pd(�P (i)): (4.15)55

Furthermore from (4.3), (4.6) and (4.12) it follows thatQi(�Q(d(x; y))) = 1ri riXj=1 vi;j(x)vi;j (y): (4.16)Thus any polynomial graph is a (distance-regular) metric space which is polyno-mial (with respect to the standard substitution �Q(d)) in the sense of the de�nition inSection 3.2. Herewith the system Q coincides with the system of orthogonal polyno-mials constructed for an arbitrary (�nite) metric space in Section 2.2. For polynomialgraphs we have one more system P of orthogonal polynomials with orthogonalityconditions (4.14). The system P also satis�es the Krein condition since from (4.2),(4.4), and (4.5) it follows that Pk;iPk;j = DXd=0 pdi;jPk;d: (4.17)The functions vd and �(d) = �Q(d) de�ne uniquely all parameters of systems Q andP: It is surprising that by a theorem of Leonard [74] only �ve of their values do that,for example, v1; v2; �(1); �(2); and �(3) (see also Terwilliger [115]). It should benoted by another important result of Leonard [73] (see also [8]) that, in fact, Q andP belong to the class of Askey-Wilson polynomials [4], which was introduced usingbasic hypergeometric series. There is also an interesting connection of Q and P withsystems of orthogonal polynomials introduced by Nikiforov and Uvarov (see [85], [86])using polynomial solutions of di�erence equations approximating (up to the secondorder) the classical hypergeometric di�erential equations on a lattice with the variablemesh �(d+ 1)� �(d).Notice the following formulae for coe�cients of expansions of an arbitrary poly-nomial f(t) =PDi=0 fi(Q)Qi(t) =PDi=0 fi(P )Pi(t) :fi(Q) = rijX j DXd=0 f(�Q(d))Qi(�Q(d))vd; (4.18)fi(P ) = vijX j DXd=0 f(�P (d))Pi(�P (d))rd: (4.19)In particular,f0(Q) = 1jX j DXd=0 f(�Q(d))vd ; f0(P ) = 1jX j DXd=0 f(�P (d))rd: (4.20)We �x a code C in a polynomial association scheme X of diameter D (or with Dclasses). For any point x 2 X we consider the vectorB(x) = (B0(x); B1(x); : : : ; BD(x)) (4.21)56

where (see (2.8)) Bi(x) = Bi(x;C) = jfy : y 2 C; d(x; y) = igj;which is called the outer distribution of C with respect to x. The vectorB = 1jCj Xx2CB(x) = (B0; B1; : : : ; BD) (4.22)is called the inner distribution of C. A code C is referred to as distance invariant ifB(x) = B(y) for any x; y 2 C (and hence B(x) = B for any x 2 C) and referred to ascompletely distance invariant if B(x) = B(y) for any x; y 2 X when d(x;C) = d(y; C)(here d(x;C) is the minimum distance between x and the points of C).For any vector a = (a0; a1; : : : ; aD) such that not all a1; : : : ; aD are equal to 0 (inparticular, for a = B and a = B(x)) we introduce the following parameters (cf. [29]) :d(a) = minfi : i = 1; : : : ; D; ai 6= 0g;s(a) = jfi : i = 1; : : : ; D; ai 6= 0gj;�(a) = � 0 if a0 = 0,1 otherwise,�(a) = � 0 if aD = 0,1 otherwise,We have denoted the values d(a); s(a); �(a) for the vector a = B by d(C); s(C);�(C). They characterize, respectively, the minimal distance, the number of (nonzero)distances and the property of a code C to be diametrical (that is, whether or not thediameter of C coincides with the diameter of the whole space X). These values forthe vector a = B(x) have a similar sense and are denoted by d(x;C); s(x;C); �(x;C)(d(B(x)) is really equal to d(x;C)). The values �(a) have an auxiliary character.They are introduced to investigate simultaneously both cases because�(B(x)) = � 1 if x 2 C,0 if x =2 C. (4.23)Now for any vector a = (a0; a1; : : : ; aD) we consider the vector a0 = (a00; a01; : : : ; a0D)de�ned by a0i = ri DXd=0 adQi(�Q(d)); i = 0; 1; : : : ; D: (4.24)The vector a0 is called theMacWilliams transform of a [78], and allows us to determinedual parameters d0(a) = d(a0); s0(a) = s(a0); �0(a) = �(a0); �0(a) = �(a0) of the57

vector a. In particular, the parameters d0(a); s0(a); �0(a) for the vector a = B playa signi�cant role for a code C as well and were denoted by d0(C); s0(C); �0(C) in thegeneral case (see (2.61), (2.62), (2.63)). The value d0(C) is called the dual distance. If�(C) is the strength of the design formed by C, that is, the maximum integer � suchthat Xx;y2CQi(�Q(d(x; y))) = 0 for i = 1; :::; �; (4.25)then from (4.24) it follows that d0(C) = �(C) + 1. The parameters d0(a); s0(a); �0(a)for a = B(x) are denoted by d0(x;C); s0(x;C); �0(x;C) respectively. Notice that�0(a) = 1 for a = B and a = B(x) for each x 2 X , (4.26)since in the both cases a00 = jCj.A polynomial f(t) is called annihilating or dual-annihilating for a = (a0; a1; : : : ; aD)(and for a code C if a = B) if respectivelyaif(�Q(i)) = 0; i = 1; : : : ; D; (4.27)a0if(�P (i)) = 0; i = 1; : : : ; D: (4.28)Annihilating and dual-annihilating polynomials for a of minimum degree (that is, s(a)and s0(a)) are called respectively minimal and dual-minimal. We denote by fC(t) andefC(t), respectively, the minimal and dual-minimal polynomial for a code C such thatfC(1) = 1 and efC(1) = 1:The following two theorems follow immediately from (4.13)-(4.15) and (4.24).Theorem 4.3. For any vector a = (a0; a1; : : : ; aD),ai = vijX j DXd=0 a0dPi(�P (d)); i = 0; 1; : : : ; D: (4.29)Theorem 4.4. For any vector a = (a0; a1; : : : ; aD) and any polynomialf(t) = DXi=0 fi(Q)Qi(t) = DXi=0 fi(P )Pi(t);the following equalities hold:DXi=0 aif(�Q(i)) = DXj=0 a0j fj(Q)rj (4.30)DXi=0 ai fi(P )vi = 1jX j DXj=0 a0jf(�P (j)): (4.31)58

Thus we introduced for a code C six parameters d(C); s(C); �(C); d0(C); s0(C);�0(C). One more parameter, the covering radius �(C) of a code C, can be de�ned asthe maximum d(a) over all a = B(x); x 2 X . A code C is called uniformly packed[12] if there exist real numbers �0; �1; : : : ; ��(C) such that�(C)Xi=0 �iBi(x) = 1 for any x 2 X . (4.32)In the next section we prove some inequalities for the parameters. The main tool isto use equalities (4.30) and (4.31) for annihilating and dual-annihilating polynomialsand also to use the adjacent (to Q and P ) systems of orthogonal polynomials describedin Section 2.3.For arbitrary a 2 f0; 1g and b 2 f0; 1g we de�ne polynomials Qa;bj (t) and P a;bj (t)in a real t of degree j; j = 0; 1; : : : ; D� �a;1 � �b;1 as follows. First we de�ne positiveconstants ca;b(Q) and ca;b(P ) by equalitiesca;b(Q) DXd=0(1� �Q(d))a(1 + �Q(d))bvd = jX j (4.33)and ca;b(P ) DXd=0(1� �P (d))a(1 + �P (d))brd = jX j; (4.34)respectively. Then the polynomials Qa;bj (t) and P a;bj (t) (together with positive con-stants ra;bj (Q) and va;bj (P )) are determined uniquely by the following orthogonalityrelationsra;bj (Q)ca;b(Q) DXd=0Qa;bi (�Q(d))Qa;bj (�Q(d))(1� �Q(d))a(1 + �Q(d))bvd = �i;j (4.35)va;bj (P )ca;b(P ) DXd=0P a;bi (�P (d))P a;bj (�P (d))(1 � �P (d))a(1 + �P (d))brd = �i;j (4.36)and normalization Qa;bj (1) = 1; P a;bj (1) = 1: (4.37)Let ta;bj (Q) and ta;bj (P ) be the largest roots of the polynomials Qa;bj (t) and P a;bj (t)respectively. Using the fact that �Q(d) and �P (d) decrease with d; we can determinevalues da;bj (Q) and da;bj (P ) as follows:�Q(da;bj (Q)) = ta;bj (Q); �P (da;bj (P )) = ta;bj (P ): (4.38)59

In particular, for the Hamming space Hnv ; d0;0k (Q) = d0;0k (P ) equals the smallest rootdk(n) of the Krawtchouk polynomial Kn;vk (z) d1(n) = v � 1v n; d2(n) = 2(v � 1)n� v + 2�p4(v � 1)n+ (v � 1)22v !and (see, for example, [71])da;bk (Q) = da;bk (P ) = dk(n� a� b) + a: (4.39)We can apply these orthogonality conditions to �nd the free coe�cient f0(Q) of thepolynomial f(t) = (1� t)a(1 + t)b (Qa;bj (t))2ta;bj (Q)� t : (4.40)Using (4.20) and the fact that by (4.35) Qa;bj (t) is orthogonal with respect to (1 ��Q(d))a(1 + �Q(d))bvd to any polynomial of degree at most j � 1, we obtainf0(Q) = 1jX j DXd=0 f(�Q(d))vd = 0: (4.41)4.2. Basic inequalities for code parameters based on annihilating polyno-mialsNow we obtain a number of inequalities for parameters of an arbitrary code C using(4.30) and (4.31) for annihilating and dual-annihilating polynomials for a = B anda = B(x). We take into account that by (4.22), (4.24) and (4.16)B0i = rijCj Xx;y2CQi(�Q(d(x; y))) = 1jCj riXj=1 Xx2C vi;j(x)!2 : (4.42)It follows that the vector B0 is nonnegative (that is, it consists of nonnegative co-ordinates). Furthermore by (4.16), (4.21) and (4.24) for the vector a = B(x) wehave B0i(x) = ri riXj=1 vi;j(x)Xy2C vi;j(y)and hence B0i = 0 implies that B0i(x) = 0 for any x 2 X . (4.43)Notice also that Bi = 0 implies that Bi(x) = 0 for any x 2 C. (4.44)60

Theorem 4.5. For any code C in a polynomial graph of diameter D the followinginequalities hold :1. d(C) + d0(C) � D + 2.2. d0(C) � 2s(C) � �(C) + 1; equality implies jCj = Q(f) where f(t) = (1 +t)��(C)(fC(t))2:3. d(C) � 2s0(C) � �0(C) + 1; equality implies jCjP (f) = jX j where f(t) =(1 + t)��0(C)( efC(t))2:4. If d0(C) � 2k � "+ 1 where k is an integer and " 2 f0; 1g, thend(C) � d1;"k�"(Q)with equality if and only if k = s(C); " = �(C) and (1 + t)"Q1;"k�"(t) is minimalfor C.5. If d(C) � 2k � "+ 1 where k is an integer and " 2 f0; 1g, thend0(C) � d1;"k�"(P )with equality if and only if k = s0(C); " = �0(C) and (1 + t)"P 1;"k�"(t) is dual-minimal for C.Remark 4.6. The �rst inequality of the theorem seems to be new although it iswell known for the Hamming and Johnson spaces and is attained for MDS-codesand Steiner systems respectively [78]. The second and third inequalities improve thecorresponding Delsarte's results when �(C) = 1 and �0(C) = 1 and are attained onlyfor tight designs and perfect codes respectively [30], [78], [67], [68], [70], [71]. Thefourth inequality follows from the author's work [64], [67] as it was noticed in [39],and is attained again for tight designs. The last inequality seems to be new and isattained only for perfect codes. It was proved in [71] for the case of the Hammingspace.A proof of Theorem 4.5 is based on some auxiliary statements on vectors a =(a0; a1; : : : ; aD).Lemma 4.7. Let g(t) be annihilating for a; �0(a) = 1 and g(�Q(i)) � 0 for i =1; 2; : : : ; D. Then d0(a) �deg g(t) + �(a): In the case �(a) = 1 equality impliesa0g(1) = a00g0(Q):Proof. One can assume that h = deg g+�(a) � D because otherwise the statementis trivial. Since a00 6= 0 and for the annihilating polynomial f(t) = (1� t)�(a)g(t) for61

a of degree at most D it holds that f0(Q) = 1jX j DXd=0 f(�Q(d))vd > 0 and a0f(1) = 0;we get from (4.30) that hXj=1 a0j fj(Q)rj 6= 0:This completes the proof of the inequality, since not all a0j ; j = 1; : : : ; h; are equalto zero, and hence d0(a) � h. The remaining part of the statement also follows from(4.30) with f(t) = g(t):Corollary 4.8. If �0(a) =1; then d0(a)+d(a) �D + 1+�(a).Proof. Use Lemma 4.7 with the polynomial g(t) =QDi=d(a)(t� �Q(i)) satisfying therequired properties.Corollary 4.9. If �0(a) =1; then d0(a) �2s(a)+�(a)��(a). In the case �(a) = 1the equality implies a0g(1) = a00g0(Q) for the polynomial g(t) = (1 + t)��(a)(f(t))2,where f(t) is minimal for a:Proof. Use Lemma 4.7 with this polynomial g(t) satisfying the required propertiesas well.Lemma 4.10. Let a be nonnegative, �0(a) =1 and d0(a) �2k � "+ �(a) where k isan integer and " 2 f0; 1g. Then d(a) �d�(a);"k�" (Q) with equality if and only if k = s(a);" = �(a); and (1 + t)�(a)Q�(a);�(a)s(a)��(a)(t) is minimal for a.Proof. Consider the polynomialf(t) = (1� t)�(a)(1 + t)" (Q�(a);"k�" (t))2t�(a);"k�" (Q)� tof degree h = 2k� "+�(a)� 1 and notice that f0(Q) = 0 by (4.41). Using (4.30) weget DXi=1 aif(�Q(i)) = 0: (4.45)It follows from Corollary 4.9 that2k � "+ �(a) � d0(a) � 2s(a) + �(a) � �(a)and hence s(a) � k. All ai; i = 1; : : : ; D, are nonnegative and exactly s(a) amongthem are positive. Since f(�Q(i)) � 0 for i � d�(a);"k�" ; it follows that (4.45) impliesd(a) � d�(a);"k�" with equality if and only if k = s(a), " = �(a) and the polynomial(1 + t)�(a)Q�(a);�(a)s(a)��(a)(t) is minimal for a. 62

Remark 4.11. By Theorems 4.3 and 4.4, Lemmas 4.7, 4.10 and Corollaries 4.8, 4.9will be valid if we replace in their formulations Q by P; a by a0, a0 by a00, and a00 bya0jX j.Proof of Theorem 4.5. We can use Corollaries 4.8, 4.9 and Lemma 4.10 for thevector a = B (see (4.22)) for which �(a) = 1 and �0(a) = 1 by (4.26). This givesthe statements 1, 2 and 4. The statements 3 and 5 follow from dual analogues ofCorollary 4.9 and Lemma 4.10 (see Remark 4.11).In connection with statement 2 and 3 of Theorem 4.5 it is also worth noting thefollowing consequences of the equalities (4.30) and (4.31).Corollary 4.12. If for a code C in a polynomial graph X there exists an annihilatingpolynomial f(t) which can be expanded over the system Q with positive coe�cients,then jCj � Q(f): Equality implies d0(C) � 1 + deg f: If for a code C there existsa dual-annihilating polynomial f(t) which can be expanded over the system P withpositive coe�cients, then jCjP (f) � jX j: Equality implies d(C) � 1 + deg f:The duality based on (4.30) and (4.31) for nonnegative vectors a = B and a0 = Band systems Q and P satisfying the Krein conditions and (4.13)-(4.15) can be appliedto prove analogs of some statements for polynomial spaces. They are obtained using(4.31) instead of (4.30). In particular, we have the following dual analog of Theorem3.17.Theorem 4.13. If for a code C in a polynomial graph X with dual-minimal poly-nomial efC(t) = s0(C)Pi=0 efiPi(t) the inequality d(C) � s0(C) + 1 holds, thenefj jX j = vj jCj; j = 0; 1; :::; d(C)� s0(C)� 1:If efC(t) can be expanded over the system P with positive coe�cients and for some d,s0(C) � d� 1 � D(X);efj jX j = vj jCj; j = 0; 1; :::; d� s0(C) � 1;then d(C) � d:For any vector a = B(x) where x =2 C; according to (4.23) and (4.26) we have�0(a) = 1; �(a) = 0. Therefore we can use only Corollaries 4.8, 4.9 and Lemma 4.10and obtain the following results for an arbitrary code C and any x 2 X nC :1. d(x;C) + d0(x;C) � D + 1;2. d0(x;C) � 2s0(x;C) � �0(x;C), 63

3. If d0(x;C) � 2k � " where k is an integer and " 2 f0; 1g thend(x;C) � d0;"k�"(Q)with equality if and only if k = s(x;C); " = �(x;C) and (1 + t)"Q0;"k�"(t) isminimal for B(x).Since by (4.43) d0(C) � d0(x;C) for any x 2 X , the last statement gives rise tothe following result.Theorem 4.14. For a code C in any polynomial graph X the inequality d0(C) �2k � " where k is an integer and " 2 f0; 1g implies that�(C) � d0;"k�"(Q)with equality if and only if there exists a point x 2 X nC such that (1 + t)"Q0;"k�"(t)is minimal for B(x).The inequality of Theorem 4.14 for the Hamming space is due to Tiet�av�ainen[117], [118]. For an arbitrary polynomial space (in particular, graph) X it was provedtogether with necessary and su�cient conditions of its attainability in [39]. In partic-ular, it is attained for all binary tight designs of even strength, for example, for thetight 6-design formed by the Golay [23,11,8] code (for this code, by Theorem 4.14,�(C) � d0;13 (23) = d3(22) = 7 since d0(C) � 7).Corollary 4.15. A code C in any polynomial graphX is distance invariant if d0(C) �s(C) or d(C) � s0(C).Proof. Given the conditions stated, we prove that the value Bi(x) for any i; i =0; 1; :::; D; does not depend on x 2 C. Since we have B0(x) = 1 for any x 2 C andby (4.44) Bi(x) = 0 for any x 2 C if Bi = 0; it is su�cient to consider the set H ofindices i 2 f1; :::; Dg such that Bi 6= 0. By the construction jH j = s(C): Accordingto (4.30) for any polynomial f(z) of degree at most d0(C)� 1;Xi2H Bi(x)f(�Q(i)) = jCjf0(Q)� f(1):On the other hand, for any j 2 H there exists the Lagrange polynomial Fj(z) ofdegree s(C) � 1 such that Fj(�Q(j)) = 1 and Fj(�Q(i)) = 0 when i 2 Hnfjg: Thiscompletes the proof if d0(C) � s(C): Analogously using (4.31) one can show that forany i; i = 0; 1; :::; D; the value B0i(x) does not depend on x 2 C if d(C) � s0(C). Thenby Theorem 4.3 it is also true for values Bi(x):Remark 4.16. Since d0(C) = �(C) + 1 the �rst statement of Corollary 4.15 andstatement 2 of Theorem 4.5 were proved, respectively, in Theorem 3.19 and Lemma64

2.17 for a code C in an arbitrary polynomial space X (in particular, for a code Con the unit Euclidean sphere). Statement 4 of Theorem 4.5 and the statement ofTheorem 4.14 are also true for a code C in an arbitrary polynomial space X as wellas Lemma 4.7, Corollary 4.9 , and Lemma 4.10 which are used in the proofs of thesestatements.We consider Bi(x) (see (4.21)) as an entry of a matrix of size jX j � (D + 1) withrows B(x) enumerated by x 2 X and columns Bi enumerated by i, i = 0; 1; : : : ; D.Theorem 4.17. For a code C in any polynomial graph X , d(x;C) � s0(C) for anyx 2 X and hence �(C) � s0(C) (4.46)with equality if and only if C is uniformly packed. The column Bs0 where s0 = s0(C)is a linear combination of the columns B0; : : : ;Bs0�1 and the column of all ones. Anycolumn Bi; s0 < i � D, is a linear combination of preceding columns, any row B(x)is uniquely determined by its �rst s0 coordinates, and the columns B0; : : : ;Bs0 arelinearly independent.Proof. Let f(t) = efC(t) be a dual-minimal polynomial of degree s0 = s0(C) for Csuch that f(1) = 1. Using (4.43) and (4.31) for a = B(x) we have for any x 2 XjX jjCj s0Xi=0 Bi(x)fi(P )vi = 1 where fs0(P ) 6= 0. (4.47)This proves the �rst two statements of the theorem apart from �(C) = s0(C) for auniformly packed code. The statement about Bi, s0 < i � D, is obtained by analogyusing the dual-annihilating polynomial ti�s0f(t) of degree i. Therefore any row B(x)is determined uniquely by its �rst s0 coordinates. The linear independence of the �rsts0 + 1 columns is a consequence of the Delsarte equalityXx2XBi(x)Bj(x) = jCj DXd=0Bdpdi;j = jCjjX j DXk=0B0kPk;iPk;jwhich follows from the de�nitions of Bi(x); Bd; pdi;j and from (4.12), (4.14), (4.15),(4.17), and shows that the rank of the matrix under consideration equals s0+1. SinceBs0 is not a linear combination of preceding columns, from (4.32) and (4.47) it followsthat for any uniformly packed code the inequality (4.46) cannot be strict.Theorem 4.17 belongs to Delsarte [30] except for the necessary and su�cient con-ditions of the equality in (4.46) which were obtained in [13]. Uniformly packed codeswere investigated in [12], [47], [13].In conclusion we derive from Theorem 4.17 two more results of Delsarte [30].Corollary 4.18. A code C in any polynomial graph X is completely distance invari-ant if d(C) � 2s0(C) � 1. 65

Proof. For any x 2 X it follows that s0�1Xi=0 Bi(x) � 1 (where s0 = s0(C)), becauseotherwise, by the triangle inequality, there exist two distinct code points at distanceat most 2s0 � 2 from each other: Furthermore by Theorem 4.17 any row B(x) isdetermined uniquely by its �rst s0 coordinates. This completes the proof.Corollary 4.19. For any code C in any polynomial graph X and s0 = s0(C);jCj � jX js0Pi=0 vi = jX jLP (t1;0s0 (P )) = jX jLP ( P (2s0 + 1)) (4.48)with equality if and only if d(C) = 2s0(C) + 1 and efC(t) = P 1;0s0 (t).Proof. According to (4.46) the metric balls of radius s0 with centers at points of acode C cover X and hence the bound (4.48) holds. If this bound is attained, thend(C) = 2s0(C) + 1 and for any i; i = 0; :::; s0; there exists x 2 X such that Bi(x) = 1:By (4.47) this means that jX jfi(P ) = jCjvi for any i; i = 0; :::; s0; and henceefC(t) = s0Xi=0 fi(P )Pi(t) = jCjjX j s0Xi=0 viPi(t) = P 1;0s0 (t):The su�ciency of the stated conditions follows from Statement 3 of Theorem 4.5.As in the case of the absolute bound of Delsarte (Theorem 3.18) one can formulatethe following conjecture.Conjecture 4.20. For any code C in a polynomial graph X;jX jLP ( P (2s0(C) � �0(C) + 1)) � jCj � jX jLP ( P (d(C))) (4.49)with equality in either of the bounds if and only ifd(C) = 2s0(C)� �0(C) + 1and efC(t) = � t+ 12 ��0(C) P 1;�0(C)s0(C)��0(C)(t):This statement is true with respect to the upper bound (it will be proved in Section5.5) and with respect to the lower bound when �0(C) = 0 (see Corollary 4.15). Thusone needs to prove only the lower bound in (4.49) when �0(C) = 1 and the fact thatthe stated conditions of its attainability are necessary in this case (the su�ciency ofthe conditions follows also from Statement 3 of Theorem 4.5). Conjecture 4.20 wascompletely proved for the Hamming space in [71].66

4.3. Duality in bounding optimal sizes of codes and designsIn Section 3.3 we considered the �-packing and � -design (continuous) problems forthe system Q which give universal bounds for codes and designs in polynomial metricspaces. Now we show that for polynomial graphs there exists a duality in bounding theoptimal sizes of codes and designs which allows us to use the corresponding extremumproblems for the system P as well.For any code C in a polynomial graph X of diameter D with the distance distri-bution B(C) = (B0(C); :::; BD(C))where Bi(C) = 1jCj jf(x; y) : x; y 2 C; d(x; y) = igj and the dual distributionB0(C) = (B00(C); :::; B0D(C))where B0i(C) = ri DXd=0Bd(C)Qi(�Q(d)) = rijCj Xx;y2CQi(�Q(d(x; y)));by Theorem 4.4 we obtainDXi=0 Bi(C)f(�Q(i)) = DXj=0B0j(C)fj(Q)rj (4.50)and DXi=0 Bi(C)fi(P )vi = 1jX j DXj=0B0j(C)f(�P (j)): (4.51)Let U = fU0(t); :::; UD(t)g be either of the systems Q or P of orthogonal polyno-mials and let f(t) = DXi=0 fi(U)Ui(t) (4.52)and U (f) = f(1)f0(U) when f0(U) 6= 0: (4.53)We say that a polynomial f(t) has the property AU (d), d = 1; :::; D + 1; iff0(U) > 0; fi(U) � 0 for i = 1; :::; D; (4.54)f(1) > 0; f(�U (i)) � 0 for i = d; :::; D; (4.55)and has the property BU (d), d = 1; :::; D + 1; iff0(U) > 0; fi(U) � 0 for i = d; :::; D; (4.56)67

f(1) > 0; f(�U (i)) � 0 for i = 1; :::; D: (4.57)Consider the following discrete extremum problems for the system U by analogywith the continuous ones.The discrete d-code problem: for any integer d; d = 1; :::; D + 1; �nd theminimum AU (d) of U (f) over all polynomials f(t) satisfying the property AU (d):The discrete (d�1)-design problem: for any integer d; d = 1; :::; D+1; �nd themaximum BU (d) of U (f) over all polynomials f(t) satisfying the property BU (d):Note that both these problems are linear programming ones because f(1) =PDi=0 fi(U) and without loss of generality one can assume that f0(U) = 1: We shallsee that the second problem for the system U is equivalent to the �rst problem forthe system U where U = P; if U = Q; and U = Q, if U = P .First, compare these discrete problems with the �-packing problem for � = �(d)and the � -design problem for � = d � 1 considered for an arbitrary system U inSection 3.3. Since the conditions (4.55) and (4.57) are weaker than, respectively,(3.78) and (3.79), any permissible solution of the continuous problem is that for thecorresponding discrete one. In particular, it follows that for any system UBU (d) � LU ( U (d)) (4.58)(see (3.104)) and, if U satis�es the strengthened Krein condition, thenAU (d) � LU (�U (d)) (4.59)(see (3.95)). Bounding A(X; d) and B(X; d) for a polynomial graph X we can use theinequalities (4.58) and (4.59) for the system U = Q and prove thatA(X; d) � AQ(d); (4.60)B(X; d) � BQ(d): (4.61)Indeed, using (4.50) and the inequalities B0j(C) � 0 we have Delsarte's result whichis analogous to Theorems 3.14 and 3.15.Theorem 4.21. For any code C in a polynomial graph X and any polynomial f(t)satisfying property AQ(d(C)) or BQ(d0(C)),jCj � Q(f) (4.62)or, respectively, jCj � Q(f): (4.63)Equality in (4.62) and (4.63) takes place if and only ifBj(C)f(�Q(j)) = 0; j = 1; :::; D; (4.64)and B0j(C)fj(Q) = 0; j = 1; :::; D: (4.65)68

Remark 4.22. To prove (4.62) and (4.63) the signs of the valuesBi(C); f(�Q(i)); B0j(C);and fj(Q) are only used in (4.50). Additional information allows us to improve thesebounds in some cases (see [78]).Note that we cannot directly use the inequalities (4.58) and (4.59) for the systemU = P because, in general, it is not true that for any code C � XXx;y2C Pi(�P (d(x; y))) � 0; i = 1; :::; D:Nevertheless, it is possible to use these inequalities by introducing a notion of Q- andP -duality of polynomials. For an arbitrary polynomial f(t); the polynomialDQ(f) = jX j� 12 DXi=0 rif(�P (i))Qi(t) (4.66)is called Q-dual to f(t) and the polynomialDP (f) = jX j� 12 DXi=0 vif(�Q(i))Pi(t) (4.67)is called P -dual to f(t): In particular, for any d = 1; :::; D consider the polynomialsm(d)Q (t) = DYi=d t� �Q(i)1� �Q(i) ; m(d)P (t) = DYi=d t� �P (i)1� �P (i) ; (4.68)of degree D � d+ 1 and note thatm(d)Q (�Q(j)) > 0 and m(d)P (�P (j)) > 0 for j = 0; 1; :::; d� 1 (4.69)under our restriction that �Q(i) and �P (i) decrease with i. LetNQ(d) = DXj=0m(d)Q (�Q(j))vj ; NP (d) = DXj=0m(d)P (�P (j))rj : (4.70)Hence NQ(d) and NP (d) are positive-valued increasing functions in d andQ(m(d)Q ) = jX jNQ(d) ; P (m(d)P ) = jX jNP (d) :Lemma 4.23. For any d; d = 1; :::; D; the polynomial jX j� 12NQ(d)m(D�d+2)P (t) is P -dual to m(d)Q (t) and belongs to F>(P ); and the polynomial jX j� 12NP (d)m(D�d+2)Q (t)is Q-dual to m(d)P (t) and belongs to F>(Q):69

Proof. According to (4.68) and (4.70) the polynomial (4.67), P -dual to f(t) =m(d)Q (t); is equal to jX j� 12NQ(d) at t = 1 and has degree d � 1 and roots �P (l);l = D � d+ 2; :::; D; since by (4.15) and (4.13),DXi=0m(d)Q (�Q(i))Pi(�P (l))vi = DXi=0m(d)Q (�Q(i))Ql(�Q(i))vi = 0for D � d + 2 � l � D: The polynomial jX j� 12NQ(d)m(D�d+2)P (t) of degree d � 1satis�es the same properties and hence is P -dual to m(d)Q (t): Its coe�cients at Pj(t);j = 0; 1; :::; d� 1; are positive by (4.69). The second part of the statement is provedanalogously.Theorem 4.24. For any polynomial f(t);DQ(DP (f)) = DP (DQ(f)) = f; (4.71)P (f)Q(DQ(f)) = Q(f)P (DP (f)) = jX j: (4.72)If f(t) satis�es property AP (d) orBP (d); then the Q-dual polynomialDQ(f) satis�es,respectively, propertyBQ(d) or AQ(d) and conversely, if f(t) satis�es property AQ(d)or BQ(d); then the P -dual polynomial DP (f) satis�es, respectively, property BP (d)or AP (d).Proof. From (4.13)-(4.15), (4.18), (4.19), and (4.52) it follows thatDXi=0 DXj=0 rivjf(�Q(j))Pj(�P (i))Qi(t) = DXi=0 DXj=0 rjvif(�P (j))Qj(�Q(i))Pi(t) = jX jf(t):In accord with de�nitions (4.66) and (4.67) this gives (4.71) and shows that the valueof DQ(f) at �Q(i) is equal to jXj 12vi fi(P ) and the value of DP (f) at �P (i) is equal tojXj 12ri fi(Q). Therefore,Q(DQ(f)) = f0(P )f(1) jX j and P (DP (f)) = f0(Q)f(1) jX jby (4.53) and (4.20), and (4.72) holds. The remaining statements are obtained if onecompares (4.54)-(4.57) with (4.66)-(4.67) and uses (4.71).Remark 4.25. From (4.28) and the proof of Theorem 4.24 condition (4.65) meansthat the P -dual polynomial DP (f) to an annihilating polynomial f(t) for a code Cis dual-annihilating for C.Corollary 4.26. For any polynomial graph X and any d; d = 1; :::; D + 1;AP (d)BQ(d) = BP (d)AQ(d) = jX j: (4.73)70

Corollary 4.26 and (4.58) allow us to state that together with the boundB(X; d) � BQ(d) � LQ( Q(d)) (4.74)valid for any polynomial space X (see Theorem 2.16 and (3.104)), for any polynomialgraph X it is also true thatA(X; d) � AQ(d) = jX jBP (d) � jX jLP ( P (d)) (4.75)(this is a generalization of the Hamming bound). Analogously, together with thebound A(X; d) � AQ(d) � LQ(�Q(d)) (4.76)valid for any polynomial space X with the system Q satisfying the strengthened Kreincondition (see (3.95)), for any polynomial graph X with the system P satisfying thestrengthened Krein condition it is also true thatB(X; d) � BQ(d) = jX jAP (d) � jX jLP (�P (d)) : (4.77)The following special cases of bounds (4.76) and (4.77) for an integer l and � 2 f0; 1gA(X; d1;�l��(Q)) � AQ(d1;�l��(Q)) � LQ(t1;�l��(Q)); (4.78)B(X; d1;�l��(P )) � BQ(d1;�l��(P )) = jX jAP (d1;�l��(P )) � jX jLP (t1;�l��(P )) (4.79)hold (see (3.101) and (4.38)) for any polynomial graph X .Thus in the case of polynomial graphs we have two pairs of universal boundsfor codes and designs, namely (4.74)-(4.75) and (4.76)-(4.77). Both these pairs areobtained using the optimal solutions of the restricted (continuous) �-packing and� -design problems (see Section 3.3) and the relationship (4.73) valid in the discretecase. There exists one more pair of universal bounds which uses essentially the discretenature of the problem considered.Theorem 4.27. For any code C in a polynomial graph X of diameter D,jX jNQ(D � d0(C) + 2) = NP (d0(C)) � jCj � jX jNQ(d(C)) = NP (D � d(C) + 2): (4.80)Each of the bounds in (4.80) is attained if and only if d(C) + d0(C) = D + 2: In thiscase m(d(C))Q (t) is annihilating and m(d0(C))P (t) is dual-annihilating for C.Proof. By Theorems 4.21 and 4.24, and Lemma 4.23 the polynomial m(d(C))Q (t) hasproperty AQ(d(C)) and gives rise to the upper bound in (4.80) which is attained71

if and only if m(d(C))Q (t) is annihilating and the P -dual (up to a constant factor)polynomial m(D�d(C)+2)P (t) is dual-annihilating for C. Moreover, if this bound isattained, then by Corollary 4.12 d0(C) � D�d(C)+2 since the polynomial m(d(C))Q (t)of degree D � d(C) + 1 belongs to F>(Q): Analogously, the polynomial m(d0(C))P (t)has property AP (d0(C)) and hence its Q-dual (up to a constant factor) polynomialm(D�d0(C)+2)Q (t) has property BQ(d0(C)) and gives rise to the lower bound in (4.80)which is attained if and only if m(D�d0(C)+2)Q (t) is annihilating and m(d0(C))P (t) is dual-annihilating for C. If the lower bound is attained, we have again d(C) � D�d0(C)+2by Corollary 4.12 since the polynomial m(d0(C))P (t) of degree D � d0(C) + 1 belongsto F>(P ): This completes the proof because NQ(d) and NP (d) increase with d andhence d0(C) � D � d(C) + 2.This theorem gives one more pair of universal bounds on A(X; d) and B(X; d) fora polynomial graph X and shows that MDS-codes and Steiner systems for which bothbounds (4.80) are attained in Hamming and Johnson space, respectively, belong tothe same class of codes with property d(C) + d0(C) = D + 2:5. Applications of orthogonal polynomials5.1. Properties of orthogonal polynomialsWhile simultaneously investigating continuous and discrete systems of orthogonalpolynomials it is convenient to use the Lebesgue-Stieltjes integral. Let �(t) be anon-decreasing left continuous function on the interval [A;B] where maybe A = �1and/or B = 1: We consider the following two cases. In the �rst case �(t) is di�er-entiable, the corresponding weight function �0(t) = w(t) is continuous on [A;B] andpositive inside the interval, and for any nonnegative integer n the integral BRA tnw(t)dtexists if A = �1 and/or B = 1: In the second case �(t) is a step function with Nsteps at points denoted by tN;i (A � tN;1 < tN;2 < � � � < tN;N � B): Assume thesteps to be wi > 0; i = 1; :::; N; and at least one of these points to be inside [A;B]:In addition suppose that the Lebesgue-Stieltjes measure � on [A;B] corresponding tothe function �(t) is normalized. In particular, these properties were assumed if �(t) ischosen to be 1� �(��1(t)) on [�1; 1] (see (2.34)) for a compact metric space. Underthese assumptions the inner product de�ned with the help of the Lebesgue-Stieltjesintegral (u; v) = BZA u(t)v(t)d�(t) (5.1)72

becomes either an integral of the typeBZA u(t)v(t)w(t)dtor represents a sum NXi=1 u(tN;i)v(tN;i)wiwhere BRA w(t)dt = 1 and NPi=1wi = 1 respectively. In the case of a discrete measure weset UN(t) = (t� tN;1)(t� tN;2) � � � (t� tN;N): (5.2)In the in�nite case we put N =1:Below we present (sometimes without proofs) the basic properties (see [14], [113])of systems of orthogonal polynomials corresponding to the inner product (5.1). Weshall normalize polynomials U(t) by U(B) = 1 if B <1 or by Co(U(t)) = 1 if B =1where Co(U(t)) is the highest degree coe�cient of the polynomial U(t) (we assumeCo(U(t)) = 1 if U(t) � 1):Theorem 5.1. There exist a unique sequence of polynomials Ui(t) of degree i; 0 �i < N; and a unique sequence of positive constants ri; 0 � i < N; such that for any iand j; 0 � i; j < N; ri BZA Ui(t)Uj(t)d�(t) = �i;j (5.3)and for any i; 0 � i < N;Ui(B) = 1 if B <1 or Co(Ui(t)) = 1 if B =1: (5.4)Note that U0(t) � 1 and r0 = 1 because of the normalization of the measure � andthat in the discrete case the polynomial UN (t) is orthogonal to all polynomials Ui(t);0 � i < N: From Theorem 5.1 it follows that for any polynomial f(t) = hPi=0 fiUi(t) ofdegree h; 0 � h < N; fi = ri BZA f(t)Ui(t)d�(t): (5.5)Theorem 5.2. Any polynomial Uk(t); 1 � k < N; has k distinct simple roots insidethe interval [A;B] : 73

Denote by tk;i (i = 1; :::; k) the roots of the polynomial Uk(t); 1 � k < N; arrangedin increasing order. In the discrete case this agrees with previous notation introducedfor the roots of the polynomial UN (t): Furthermore, we denote by tk the largest zerotk;k of the polynomial Uk(t): Note that by Theorem 5.2 and the normalization (5.4)the highest coe�cients of polynomials Uk(t) are positive and, hence,sgn Uk(A) = (�1)k for 0 � k < N . (5.6)Since (tUi(t); Uj(t)) = (Ui(t); tUj(t)) = 0 for any j; 0 � j � i� 2; we haveTheorem 5.3 (recurrence). For any i; 0 � i < N; there exist real numbers ai; bi;and ci (c0 = 0) such that(t� ai)Ui(t) = biUi+1(t) + ciUi�1(t): (5.7)We can see that bi = Co(Ui(t))Co(Ui+1(t)) ; (5.8)ci = (Ui(t); tUi�1(t))(Ui�1(t); Ui�1(t)) = ri�1ri bi�1 (5.9)and, hence, all numbers bi; i � 0; and ci; i � 1; are positive in accord with the chosennormalization (5.4).Theorem 5.4 (Christo�el-Darboux formulae). For any k; 0 � k < N; and anyreal numbers x and y;kXi=0 riUi(x)Ui(y) = rkbkUk+1(x)Uk(y)� Uk(x)Uk+1(y)x� y if x 6= y; (5.10)kXi=0 riUi(x)Ui(x) = rkbk(U 0k+1(x)Uk(x)� U 0k(x)Uk+1(x)): (5.11)Corollary 5.5 (monotonicity). For any k; 0 � k < N;Uk+1(t)Uk(t) increases with t if Uk(t) 6= 0: (5.12)Corollary 5.6 (separation of roots). For any k and j such that 1 � j � 1 �k � 1 < N; tk;j�1 < tk�1;j�1 < tk;j : (5.13)Corollary 5.7. If tk�1 < t < tk; 0 � k � 1 < N; thenUk(t) < 0 and Ui(t) > 0 for any i; i = 0; 1; :::; k � 1:If t � tk; then Uk(t) � 0: 74

It is convenient to introduce a special notation for the left-hand side of (5.10). For0 � k < N let Tk(x; y) = kXi=0 riUi(x)Ui(y): (5.14)For a polynomial g(t) of degree k having k simple roots x1; :::; xk denote by li(g; t);i = 1; :::; k; the Lagrange polynomials of degree k � 1 such that li(g;xj) = �i;j : It isclear that li(g; t) = g(t)(t� xi)g0(xi) : (5.15)In particular, from (5.10) and (5.11) it follows that for 1 � k < N and for k = N ifN <1; li(Uk; t) = Tk�1(t; tk;i)Tk�1(tk;i; tk;i) : (5.16)One of fruitful ideas of the theory of orthogonal polynomials consists in the fact thatfor any polynomial f(t) of degree at most 2k � 1; 1 � k < N; the polynomialf(t)� kXi=1 f(tk;i)li(Uk; t) (5.17)is divisible by Uk(t) and, hence, orthogonal to U0(t) � 1: In particular, this gives thefollowing well-known theorem.Theorem 5.8 (Gauss-Jacobi formula). For any polynomial f(t) of degree at most2k � 1; 1 � k < N; f0 = BZA f(t)d�(t) = kXi=1 f(tk;i)�k;iwhere �k;i = BZA li(Uk; t)d�(t) = 1Tk�1(tk;i; tk;i) > 0:Note that for discrete systems the orthogonality condition (5.3) takes the formri NXl=1 Ui(tN;l)Uj(tN;l)wl = �i;j (5.18)where NPl=1wl = 1: Since li(UN ; tN;l) = �i;l and NPl=1Ui(tN;l)wl = �i;0; from (5.16) wherek = N; it follows that for any i; i = 1; 2; :::; N;wi = NXl=1 li(UN ; tN;l)wl = 1TN�1(tN;i; tN;i)75

and, hence, wi N�1Xl=0 Ul(t)Ul(tN;i)rl = li(UN ; t):In particular, we have the following statement.Lemma 5.9 (the second orthogonality condition for discrete systems). IfN <1; then for any i and j; 1 � i; j � N;wi N�1Xl=0 Ul(tN;i)Ul(tN;j)rl = �i;j : (5.19)If there exists a substitution function �(l) such that for any i; i = 1; :::; N; thevalue Ul(tN;i) is a polynomial of degree i�1 in �(l); then (5.19) gives the orthogonalitycondition for these polynomials. In particular, this holds for polynomial graphs (see(4.13)-(4.15)).The next lemma follows from (5.14), (5.3), and (5.10).Lemma 5.10. For any k and l, 0 � k; l < N;BZA Tk(t; x)Tl(t; y)d�(t) = Tmin(k;l)(x; y); (5.20)BZA (t� x)Tk(t; x)Tl(t; y)d�(t) = ��k;lrkbkUk(x)Uk+1(y); (5.21)and if x and y are distinct roots of the equation (t��)Tk(t; �) = 0 for a �xed �, thenTk(x; y) = 0: (5.22)We further introduce the functionRk(x; y; z) = Tk�1(x; y)Uk(z)� Tk�1(y; z)Uk(x) (5.23)for 1 � k < N and note that because of (5.14)Rk(x; y; z) = Tk(x; y)Uk(z)� Tk(y; z)Uk(x): (5.24)From (5.10), (5.14), and (5.23) it follows that for 1 � k < N(z � y)Rk(x; y; z) = (z � x)Rk(y; x; z) (5.25)and (y � x)(z � y)Rk(x; y; z) (rk�1bk�1)�1 = �(z � x)Uk�1(y)Uk(x)Uk(z)76

+Uk(y) ((z � y)Uk�1(x)Uk(z) + (y � x)Uk(x)Uk�1(z)) : (5.26)On the other hand, from (5.10), (5.14), and (5.24) it follows that for 1 � k < N(y � x)(z � y)Rk(x; y; z) (rkbk)�1 = (z � x)Uk+1(y)Uk(x)Uk(z)�Uk(y) ((z � y)Uk+1(x)Uk(z) + (y � x)Uk(x)Uk+1(z)) : (5.27)Using (5.3), (5.20), (5.23), (5.24) and then the equalities(z � t)Rk(x; t; z) = (z � x)Rk(t; x; z);(t� x)Rk(x; t; z) = (x� t)Rk(z; t; x) = (x� z)Rk(t; z; x);we get we following statement.Lemma 5.11. For any k and l (1 � k; l < N);rk BZA Rk(t; x; z)Rl(t; z; x)d�(t) = �k;lTk�1(x; z)Tk(x; z); (5.28)rk BZA (t�x)(z� t)Rk(x; t; z)Rl(x; t; z)d�(t) = ��k;l(x� z)2Tk�1(x; z)Tk(x; z): (5.29)5.2. Bounds on extreme roots of orthogonal polynomialsTheorems 5.3 and 5.8 allow us to give the following useful expressions for the extreme(the smallest and the largest) roots tk;1 and tk;k = tk of the polynomials Uk(t) ofdegree k which form the system of orthogonal polynomials considered. Note thatt1 = a0:Theorem 5.12. For any k, 1 � k < N;tk;1 = min k�1Xi=0 aix2i � 2 k�2Xi=0 xixi+1pbici+1! , (5.30)tk;k = max k�1Xi=0 aix2i + 2 k�2Xi=0 xixi+1pbici+1! , (5.31)where the extrema are taken over all (or only over positive) x0; x1; : : : ; xk�1 such thatk�1Xi=0 x2i = 1: (5.32)77

Proof. By (5.4), (5.6), and (5.13) for any i and k; 0 � i < k < N;(�1)iUi(tk;1) > 0 and Ui(tk;k) > 0:To prove (5.30) we consider the polynomial f(t) = (t � tk;1)(g(t))2 of degree 2k � 1;where g(t) = k�1Xi=0 xipri (�1)i Ui(t)and xi; i = 0; 1; :::; k � 1; satisfy (5.32). Using (5.7) and (5.9) we havef0 = BZA f(t)d�(t) = BZA (t� tk;1) (g(t))2 d�(t)= BZA g(t) k�1Xi=0 xipri (�1)i (biUi+1(t) + (ai � tk;1)Ui(t) + ciUi�1(t))! d�(t)= k�1Xi=0 aix2i � k�1Xi=1 xixi�1rri�1ri bi�1 � k�2Xi=0 xixi+1rri+1ri ci+1 � tk;1 k�1Xi=0 x2i= k�1Xi=0 aix2i � 2 k�2Xi=0 xixi+1pbici+1 � tk;1:Note that in the special case whenxi = pri (�1)i Ui(tk;1)pTk�1(tk;1; tk;1) ; i = 0; 1; :::; k � 1; (5.33)(and all xi > 0) the equality (5.32) holds and we haveg(t) = Tk�1(t; tk;1)pTk�1(tk;1; tk;1)and, hence, by (5.16) f(tk;i) = 0 for all i = 1; :::; k: Using Theorem 5.8 for thepolynomial f(t) = (t � tk;1)(g(t))2 we get f0 � 0 with equality in the special case(5.33). This completes the proof of (5.30). Analogously considering the polynomialf(t) = (tk;k � t)(g(t))2 of degree 2k � 1; whereg(t) = k�1Xi=0 xipriUi(t)and xi; i = 0; 1; :::; k� 1; satisfy (5.32), we prove that (5.31) holds and the maximumis attained when xi = priUi(tk;k)pTk�1(tk;k ; tk;k) ; i = 0; 1; :::; k � 1: (5.34)78

Remark 5.13. Though (5.33) and (5.34) are solutions of these extremum problemsthey are expressed by functions in values we would like to �nd. Nevertheless relation-ships (5.30) and (5.31) turn out to be very useful for bounding extreme roots with thehelp of the known extrema of some simple quadratic forms and the monotonicity ofsome sequences. The author has not found this elementary and general statement inthe literature on orthogonal polynomials (cf. [56]) and published it with applicationsto coding theory in [71].Remark 5.14. As should be expected Theorem 5.12 is invariant with respect to achoice of normalization (5.4) of polynomials Uk(t) and the measure �. Indeed, ifeU = neUk(t); 0 � k < No where eUk(t) = dkUk(t) and dk 6= 0; then the roots of eUk(t)and Uk(t) coincide and coe�cients eai; ebi; eci of the recurrence for the system eU areconnected with those ai; bi; ci for the system U (see (5.7)) as followseai = ai; ebi = bidi=di+1; eci = cidi=di�1 and, hence, ebieci+1 = bici+1:Thus, applying Theorem 5.12 we can use any normalization of polynomials and mea-sure.Consider the Hermite polynomials Pk(t) of degree k; k = 0; 1; : : : , which areorthogonal on (�1;1) with the respect to the weight function exp(�t2) and whichcan be de�ned by the following recurrence:Pk+1(t)� 2tPk(t) + 2kPk�1(t) = 0; P0(t) = 1(see [14], [113]). Then the largest root hk of Pk(t); k = 1; : : : , by Theorem 5.12 canbe represented as follows hk = maxp2 k�2Xi=0 xixi+1pi+ 1; (5.35)where the maximum is taken over positive x0; x1; : : : ; xk�1 satisfying (5.32). This rootand, hence, the maximum of the corresponding quadratic form were calculated withhigh accuracy. It is known that h1 = 0; p2h2 = 1; p2h3 = p3; p2h4 = p3 +p6,p2h5 =p5 +p10; and (see [113])hk = p2k�c(2k)�1=6 + o(k�1=6) as k !1 (5.36)where c = 1:85575 : : : : We can use Theorem 5.12 and these bounds on the quadraticform (5.35) for estimating the extreme roots of other systems of orthogonal polynomi-als. We formulate the following statement only for the largest roots; the correspondingstatement for the smallest roots is also valid.79

Lemma 5.15. Given k; 2 � k < N; suppose that ai � ai+1; i = 0; :::; k � 2; andthere exists a sequence mi; i = 1; :::; k� 1; such that ai +pbi�1ci +pbi�2ci�1 � miand mi � mi+1. Then tk;k � mk�1: (5.37)If additionally to these properties, mi � ai+1+pbici+1+pbi�1ci+C, where C doesnot depend on i, then for any l (l = 2; :::; k),tk;k � l � 2l (mk�l+1 � C) + 2l ak�l: (5.38)Proof. Using (5.31), the inequality 2xixi+1 � x2i + x2i+1; and the monotonicity of aiand mi we havetk;k � k�2Xi=0 x2imi+1 + x2k�1 �ak�1 +pbk�2ck�1� � mk�1:If we put xk�1 = ::: = xk�l = l�1=2 in (5.31), we gettk;k � 1l k�1Xi=k�l ai + 2 k�2Xi=k�lpbici+1! (5.39)and use the motonicity of ai and mi and the upper bound for mi to obtain (5.38).We shall see that the proper choice of l = l(k) in Lemma 5.15 can prove thatthe inequalities (5.37) are asymptotically tight. In particular, this lemma for mi =q i2 +q i�12 shows that p2hk � pk � 1 + pk � 2 and (put l = l(k) = jpkk) thathk = p2k +O(1) as k !1 (cf. (5.36)).Apply Theorem 5.12 and Lemma 5.15 to the Jacobi polynomials P�;�k (t) of degreek = 1; 2; ::: ; de�ned by (2.48) and orthogonal on [�1; 1] with respect to weight functionc�;�(1� t)�(1+ t)� where the normalizing constant c�;� is de�ned by (2.50). Let p�;�kbe the largest root of P�;�k (t): In particular,p�;�1 = � � ��+ � + 2 ; p�;�2 = 2p(�+ � + 3)(2 + �)(2 + �)� (�+ � + 3)(�� �)(�+ � + 3)(�+ � + 4) :For these polynomials the recurrence (5.7) holds for the following coe�cients:ak = �2 � �2(2k + �+ �)(2k + �+ � + 2) , bk = 2(k + �+ 1)(k + �+ � + 1)(2k + �+ � + 1)(2k + �+ � + 2) ,ck = 2k(k + �)(2k + �+ �)(2k + �+ � + 1) :One can verify that ai+1 � ai, if � � �, and bici+1 � bi�1ci, if �+ � � 1 (otherwise,in general, the latter is not true). Therefore the conditions of Lemma 5.15 are ful�lledfor mi = ai+2pbi�1ci (and for mi = ai+pbi�1ci+pbi�2ci�1 as well) with C = 0.80

Corollary 5.16. If � � � � � 12 and �+ � � 1; then for every k; k = 2; : : : ,p�;�k � �2 � �2(2k + �+ � � 2)(2k + �+ �)+4s (k � 1) (k + �� 1)(k + � � 1)(k + �+ � � 1)(2k + �+ � � 3)(2k + �+ � � 2)2(2k + �+ � � 1) :Moreover, if limk!1 �k = � and limk!1 �k = �, where � � � � 0, thenp�;�k = 4p(1 + �)(1 + �)(1 + � + �)� �2 + �2(2 + � + �)2 + o(1): (5.40)The asymptotic formula (5.40) was obtained in [57] with the help of Sturm-Liouville arguments.In the case � = � we can also use (5.35) and the fact that bici+1=(i+1) decreaseswith i to obtain a lower bound.Corollary 5.17. If � = � � 12 ; then for every k; k = 2; : : : ,p2hk � p�;�k r (2k + 2�� 3)(2k + 2�� 1)k + 2�� 1 � 2pk � 1:Considering systems Q and P for polynomial graphs and using the fact that �Q(d)and �P (d) decrease with d; we de�ne values dk(Q) and dk(P ) by �Q(dk(Q)) = tk(Q)and �P (dk(P )) = tk(P ) (cf. (4.38)). If �Q(d) (respectively �P (d)) is a linear function,then dk(Q) (respectively, dk(P )) is the smallest root of a polynomial of degree k: Inparticular, this is true for Hamming and Johnson spaces with respect to the systemQ and the corresponding polynomials are Krawtchouk and Hahn polynomials. Notethe following useful restrictions which follow from equalities Qk(�Q(j)) = Pj(�P (k))(see (4.15)) and monotonicity of �Q(d) and �P (d).Lemma 5.18. For a polynomial graph,dk(Q) � 1 if k � d1(P ); (5.41)dk(P ) � 1 if k � d1(Q): (5.42)Proof. If k � d1(P ); then �P (k) � �P (d1(P )) = t1(P ) and hence Qk(�Q(1)) =P1(�P (k)) � 0. Therefore �Q(1) � tk(Q) = �Q(dk(Q)) and hence (5.41) holds.Analogously (5.42) is proved.For the Hamming space, P = Q and dk(P ) = dk(Q) is the smallest root dk(n) =dk(n; v) of the Krawtchouk polynomialKnk (z) = Kn;vk (z) of degree k de�ned by (2.44).Since d1(n) = v�1v n; (5.41) means that dk(Q) � 1 if k � v�1v n:We use Theorem 5.12,Remark 5.14 and the known recurrence(k + 1)Knk+1(z) = ((v � 1)n� (v � 2)k � vz)Knk (z)� (v � 1)(n� k + 1)Knk�1(z):81

Corollary 5.19. For every k; k = 1; : : : ; n,dk(n) = v � 1v n� 1v max (v � 2) k�1Xi=0 ix2i + 2pv � 1 k�2Xi=0 xixi+1p(i+ 1)(n� i)!(5.43)where the maximum is taken over positive x0; x1; : : : ; xk�1 satisfying (5.32).Since �Pk�2i=0 xixi+1pi+ 1�2 �Pk�2i=0 x2i Pk�1i=1 ix2i , using (5.35) we haveCorollary 5.20. For every k; k = 1; : : : ; n,dk(n) � v � 1v n� v � 22v h2k � 1vp2(v � 1)(n� k + 2)hk: (5.44)On the other hand, from Corollary 5.19 and (5.35) it follows that for any k;k = 1; : : : ; n, vdk(n) � (v � 1)n� (v � 2)(k � 1)�p2(v � 1)nhk: (5.45)It is also true that for any k; 2 � k � n(v � 1)=v,vdk(n) � (v � 1)n� (v � 2)(k � 1)�pv � 1�p(k � 1)(n� k + 2) +p(k � 2)(n� k + 3)� (5.46)(to estimate from above the maximum in (5.43) one can use, as in the proof of Lemma5.15, thatmi � mi+1 formi = (v�2)i+pv � 1�pi(n� i+ 1) +p(i� 1)(n� i+ 2)�,when i = 1; :::; k � 2).By virtue of (5.36), (5.44), and (5.46) the following holds.Corollary 5.21. For n!1 and any k = k(n), 1 � k � n(v � 1)=v,dk(n)n = v(kn ) + o(1); (5.47)where v(x) = 1v �v � 1� (v � 2)x� 2p(v � 1)x(1� x)� : (5.48)For the Johnson space the system fQi(t); i = 0; 1; :::; wg; de�ned by (2.47) with thehelp of Hahn polynomials Ji(z) = Jn;wi (z) (see (2.46)), satis�es (5.7) with coe�cientsai = (n� 2w)(2i(n� i+ 1)� w(n+ 2))w(n� 2i)(n� 2i+ 2) , (5.49)bi = 2(w � i)(n� w � i)(n� i+ 1)w(n� 2i)(n� 2i+ 1) , ci = 2i(w � i+ 1)(n� w � i+ 1)w(n� 2i+ 1)(n� 2i+ 2) :82

The largest root tk(Q) of Qk(t) and the smallest root dk(Q) of Ji(z) are connected asfollows: tk(Q) = 1� 2wdk(Q): (5.50)Since t1(P ) = �n�w�1n�w+1 and d1(P ) = 12 �n+ 1�p(n+ 1)2 � 4w(n� w)�(see a0 be-low in (5.56) and (5.55)), by Lemma 5.18, dk(Q) � 1 if k � d1(P ), and, hence, wecan consider k < 12 �n+ 1�p(n+ 1)2 � 4w(n� w)� < w: It is clear that ai � ai+1;when 0 � i � k � 2. For i = 1; :::; k � 1; de�nemi = 1� 2 (w(n� w) � i(n� i))w �n+ 2pi(n� i)� + C with C = 6pk(n� k)w(n� 2k) . (5.51)One can check that ai+pbi�1ci+pbi�2ci�1 � mi � ai+1+pbici+1+pbi�1ci+C.Moreover, mi � mi+1; 1 � i � k � 2, because the derivative with respect to x ofthe function x��(1��)1+2px , where x = i(n�i)n2 and � = wn , is nonnegative in the range0 � x � �(1 � �); 0 < � � 12 . Therefore, for any k, 2 � k < w, the conditionsof Lemma 5.15 are ful�lled for the sequence mi and the value C de�ned by (5.51).In particular, this proves that tk(Q) � mk�1 and gives the following asymptoticequalities for tk(Q) and dk(Q) (see (5.50)).Corollary 5.22. If limn!1 kn = � and limn!1 wn = �; where 0 � � � � and 0 < � � 12 ;then tk(Q) = 1� 2 (�(1� �)� �(1� �))�(1 + 2p�(1� �)) + o(1), (5.52)dk(Q)n = �(1� �)� �(1� �)(1 + 2p�(1� �)) + o(1): (5.53)We shall use below that for 0 � i � k � 1 � w � 2 the coe�cients (5.49) have thefollowing properties: ci+1 � ci � 2(w � i)nw(n� 2i)(n� 2i� 1) ;pbici+1 �pbi+1ci+1 � (n� w � i)nw(n� 2i)(n� 2i� 1) ;and, hence, providedPk�1i=0 x2i = 1;k�1Xi=0 x2i (ci+1 � ci) + 2 k�2Xi=0 xixi+1 �pbici+1 �pbi+1ci+1� � 2nw(n� 2k + 1) . (5.54)83

In conclusion we consider the largest zeros tk(P ) = tk(n;w) of the polynomialsPk(t) = Pn;wk (t) de�ned for the Johnson space (see Example 4.2 and (4.12)) and thecorresponding values dk(P ) such thattk(P ) = 1� 2dk(P )(n+ 1� dk(P ))w(n + 1� w) : (5.55)For the polynomials Pk(t) the recurrence (5.7) holds withai = 2in� w(n� w � 1)� 4i2w(n + 1� w) ; bi = 2(w � i)(n� w � i)w(n+ 1� w) ; ci = 2i2w(n + 1� w) :(5.56)Since d1(Q) = w(n�w)n (see a0 in (5.49) and (5.49)), by Lemma 5.18, dk(P ) � 1 whenk � w(n�w)n ; and hence we can consider k < w(n�w)n � n4 : We again have ai � ai+1;when 0 � i � k � 2. For i = 1; :::; k � 1; de�nemi = 1� 2�pw(n� w) � i(n� i)� i�2w(n� w + 1) + C with C = 3w . (5.57)One can check that ai+pbi�1ci+pbi�2ci�1 � mi � ai+1+pbici+1+pbi�1ci+C.Moreover, mi � mi+1; 1 � i � k � 2, because the derivative with respect to x ofthe function ��p� � x(1� x)� x�2, where x = in and � = w(n�w)n2 , is nonnegativein the range 0 � x � � � 14 . Therefore, for any k, 2 � k < w(n�w)n , the conditionsof Lemma 5.15 are ful�lled for the sequence mi and the value C de�ned by (5.57).In particular, this proves that tk(P ) � mk�1 and gives the following asymptoticequalities for tk(P ) and dk(P ) (see (5.55)).Corollary 5.23. If limn!1 kn = � and limk!1 wn = �; where 0 � � � �(1 � �) and0 < � � 12 ; then tk(P ) = 1� 2�p�(1� �)� �(1� �)� ��2�(1� �) + o(1); (5.58)dk(P )n = 12 1�r1� 4�p�(1� �)� �(1� �)� ��2!+ o(1): (5.59)5.3. Properties of adjacent systems of orthogonal polynomialsWe continue investigation of systems U = fUk(t); 0 � k < Ng of orthogonal poly-nomials introduced in Section 5.1. However now we shall assume that they areorthogonal on the interval [�1; 1] (i.e., A = �1; B = 1) with the normalization84

Uk(1) = 1 (see (5.4)). For this normalization by Theorem 5.1 any system U and thecorresponding system of positive constants ri are uniquely determined by the weightfunction w(t) in the continuous case or by the steps wi at the points tN;i; i = 1; :::; N;in the discrete case. For any nonnegative integers a and b we can use the fact todetermine adjacent (to U) systems Ua;b = nUa;bk (t); 0 � k < Na;bo of orthogonalpolynomials (and positive constants ra;bi ) considering, respectively, weight functionswa;b(t) = ca;b(1 � t)a(1 + t)bw(t) or steps wa;bi = ca;b(1 � tN;i)a(1 + tN;i)bwi at thesame points tN;i; i = 1; :::; N; where the positive constants ca;b are chosen as followsca;b 1Z�1 (1� t)a(1 + t)bw(t)dt = 1; ca;b NXi=1(1� tN;i)a(1 + tN;i)bwi = 1 (5.60)to normalize the corresponding measure. Thus the orthogonality and normalizationconditions for the system Ua;b may be rewritten in the following formra;bi 1Z�1 Ua;bi (t)Ua;bj (t)ca;b(1� t)a(1 + t)bd�(t) = �i;j ; Ua;bi (1) = 1: (5.61)In particular, the systems of Jacobi polynomials with integer parameters are adjacentto the system of Legendre polynomials (with �(t) = t).Our immediate goal is to establish a connection between the parameters of theadjacent systems Ua;b of orthogonal polynomials and those of the original system U .All the content of Sections 5.1 and 5.2 remains valid if we replace the parameters w(t);wi; Uk(t); N; rk; ak; bk; ck; tk;i; tk; Tk(x; y); Rk(x; y; z) by the corresponding parame-ters wa;b(t); wa;bi ; Ua;bk (t); Na;b; ra;bk ; aa;bk ; ba;bk ; ca;bk ; ta;bk;i ; ta;bk ; T a;bk (x; y); Ra;bk (x; y; z).Note that Ua;b0 (t) � 1 and ra;b0 = 1; the same as for the system U . In the discrete casethe number Na;b of positive steps may be less than N . In particular, if tN;1 = �1and tN;N = 1; then Na;b = N � 2 + �a;0 + �b;0.We shall often take into account that by (5.6), (5.10) and (5.23),sgn Tk(1;�1) = (�1)k for 0 � k < N0;1 (5.62)(TN�1(1;�1) = 0 if tN;N = 1 and tN;1 = �1; however, N0;1 = N �1 in this case) andsgn Rk+1(�1; 1; 1) = (�1)k for 1 � k + 1 < N . (5.63)In particular, using (5.61)-(5.63), Theorem 5.1 and Lemmas 5.10 and 5.11 we havethe following statement.Lemma 5.24. For any k (0 � k < Na;b; respectively)U0;1k (t) = Tk(t;�1)Tk(1;�1) ; r0;1k = (Tk(1;�1))2�c0;1rkbkUk(�1)Uk+1(�1) ; (5.64)85

U1;0k (t) = Tk(t; 1)Tk(1; 1) ; r1;0k = (Tk(1; 1))2c1;0rkbkUk(1)Uk+1(1) ; (5.65)U1;1k (t) = Rk+1(�1; t; 1)Rk+1(�1; 1; 1) ; r1;1k = rk+1 (Rk+1(�1; 1; 1))2�4c1;1Tk(1;�1)Tk+1(1;�1) : (5.66)If in the discrete case N1;1 = N � 2 and, hence, tN;N = 1 and tN;1 = �1; thenU1;1N�2(t)U1;1N�2(1) = RN�1(�1; t; 1)RN�1(�1; 1; 1) = (t� tN;2) � � � (t� tN;N�1)(1� tN;2) � � � (1� tN;N�1) :Now we deduce some consequences of Lemma 5.24 and previous results.Corollary 5.25. For any k, 0 � k < N1;1;U1;1k (t) = lkU0;1k (t) +mkU1;0k (t)wherelk = Uk+1(1)Tk(�1; 1)Rk+1(�1; 1; 1) U1;1k (1) > 0 and mk = �Uk+1(�1)Tk(1; 1)Rk+1(�1; 1; 1) U1;1k (1) > 0:In the discrete case this is also true for k = N1;1 when N1;1 = N � 2:Lemma 5.24 and (5.8), (5.9), (5.61) allow us to express the coe�cients aa;bk ; ba;bk ;ca;bk via parameters of the original system U . In particular,b1;0k = ck+1 Tk+1(1; 1)Tk(1; 1) ; c1;0k = bk Tk�1(1; 1)Tk(1; 1) ; b1;0k c1;0k+1 = bk+1ck+1;a1;0k = 1� b1;0k � c1;0k = 1� bk � ck+1 = ak + ck � ck+1:Using Theorem 5.15 we have the following.Corollary 5.26. For any k, 2 � k + 1 < N;t1;0k = max k�1Xi=0 (ai + ci � ci+1) x2i + 2 k�2Xi=0 xixi+1pbi+1ci+1! ,where the maximum is taken over all x0; x1; : : : ; xk�1 such that Pk�1i=0 x2i = 1:Remark 5.27. Any adjacent system of orthogonal polynomialsUa;b = nUa;bk (t); 0 � k < Na;bocan serve as the original system U and thus the results of Sections 5.1 and 5.2 maybe applied to this system. In particular, Lemma 5.24 enables us to express the poly-nomials Ua+1;bk (t) and Ua;b+1k (t) in terms of polynomials Ua;bk (t). For example,U0;2k (t) = T 0;1k (t;�1)T 0;1k (1;�1) ; U2;0k (t) = T 1;0k (t; 1)T 1;0k (1; 1) ; (5.67)86

U1;1k (t) = T 1;0k (t;�1)T 1;0k (1;�1) = T 0;1k (t; 1)T 0;1k (1; 1) : (5.68)The next lemma follows from this remark, Lemma 5.24, (5.10), (5.14), (5.23), and(5.24).Lemma 5.28. For any k ; 2 � k + 1 < N; and any real x and �(t� �)T 0;1k�1(t; �)(x � �)T 0;1k�1(x; �) = Uk(t)Tk�1(�;�1)� Uk(�)Tk�1(t;�1)Uk(x)Tk�1(�;�1)� Uk(�)Tk�1(x;�1) ; (5.69)(t� �)T 1;0k�1(t; �)(x� �)T 1;0k�1(x; �) = Uk(t)Tk�1(�; 1)� Uk(�)Tk�1(t; 1)Uk(x)Tk�1(�; 1)� Uk(�)Tk�1(x; 1) ; (5.70)(t� �)T 1;1k�1(t; �)(x � �)T 1;1k�1(x; �) = Tk(t; 1)Tk(�;�1)� Tk(�; 1)Tk(t;�1)Tk(x; 1)Tk(�;�1)� Tk(�; 1)Tk(x;�1) ; (5.71)when the denominators in expressions (5.69)-(5.71) di�er from zero.The following two statements are based on analysis of the equations (1+t)U0;1k (t) =0; (1 � t2)U1;1k�1(t) = 0; and (1 � t)U1;0k (t) = 0 by using Lemma 5.24, Theorems 5.2and 5.4, Corollaries 5.5 and 5.6, and the equalities (5.27) and (5.26), respectively.Lemma 5.29. For any j and k such that 1 � j � k + 1 < N;tk;j�1 < t0;1k;j�1 < t1;1k�1;j�1 < t1;0k;j < tk;j (5.72)in the cases when the corresponding entries in (5.72) are de�ned.Lemma 5.30. For any j and k such that 1 � j � k � 1; 2 � k < N;tk;j < t1;0k�1;j < t1;1k�1;j < t0;1k�1;j < tk;j+1: (5.73)These lemmas on the separation of roots of adjacent polynomials can be used toprove some results on the monotonicity of the ratio of adjacent polynomials (see [68]).Lemma 5.31. The functionsUk(t)U1;0k (t) ; U0;1k (t)Uk(t) ; U0;1k (t)U1;0k (t)increase in t in those cases where denominators di�er from zero.Lemma 5.32. The functionsU0;1k�1(t)Uk(t) ; U1;1k�1(t)Uk(t) ; U1;0k�1(t)Uk(t)decrease in t if Uk(t) 6= 0. 87

One more statement on separation of roots follows from analysis of the equations(1 + t)U0;2k�1(t) = 0 and (1� t)U2;0k�1(t) = 0 by using (5.67) and Lemma 5.32.Lemma 5.33. For any j and k such that 1 � j � k � 1; 2 � k < N;tk;j < t2;0k�1;j < t1;0k�1;j < t0;1k�1;j < t0;2k�1;j < tk;j+1: (5.74)From the equalities (5.23), (5.24), and (5.68)-(5.70) we obtain the following state-ments.Lemma 5.34. For any k, 2 � k < N; the equation (t � 1)U1;1k�1(t) = 0 is equivalentto either of the equationsU0;1k�1(t) = Uk(t) and U0;1k (t) = Uk(t);and the equation (t+ 1)U1;1k�1(t) = 0 is equivalent to either of the equationsU1;0k�1(t)Uk(�1) = Uk(t)U1;0k�1(�1) and U1;0k (t)Uk(�1) = Uk(t)U1;0k (�1):Corollary 5.35. For any k, 2 � k < N;Uk(t1;1k�1) = U0;1k�1(t1;1k�1) = U0;1k (t1;1k�1)= U1;0k�1(t1;1k�1) Uk(�1)U1;0k�1(�1) = U1;0k (t1;1k�1) Uk(�1)U1;0k (�1) :Corollary 5.36.(t� �)T 1;0k�1(t; �)(1� �)T 1;0k�1(1; �) = � t+12 U1;1k�1(t) if � = t1;1k�1; 1 � k < N1;1;U1;0k (t) if � = t1;0k ; 1 � k < N1;0; (5.75)(t� �)T 1;1k�1(t; �)(1� �)T 1;1k�1(1; �) = � U1;0k (t) if � = t1;0k ; 1 � k < N1;0;U1;1k (t) if � = t1;1k ; 1 � k < N1;1: (5.76)5.4. Main theorem and consequencesNow for a system U of orthogonal polynomials we consider extremum problems ofessential interest in coding and design theory of which solutions are expressed inparameters of adjacent systems Ua;b: In the discrete case we shall assume additionallythat tN;N = 1 (and, hence, N1;0 = N � 1; N1;1 = N0;1 � 1). This is ful�lled forthe systems Q (and P ) for �nite polynomial spaces with standard substitutions (seeSection 2.1 and 5.1). We denoted by F [t] the set of polynomials in real t with real88

coe�cients. Any polynomial f 2 F [t] can be uniquely represented (mod UN(t) in thediscrete case, see (5.2)) in the form N�1Pi=0 fiUi(t) wherefi = fi(U) = ri 1Z�1 f(t)Ui(t)d�(t): (5.77)We also introduced the notation(f) = U (f) = f(1)f0(U) when f0(U) 6= 0: (5.78)(Note that in the discrete case polynomials which are equivalent modulo UN (t) havethe same values f(1); f0(U) and, hence, U (f) because of tN;N = 1). First for any�; �1 � � < 1; we study the problem of �nding the in�mum of U (f) over the classof polynomials f(t) 2 F [t] such that f0(U) > 0 andf(t) � 0 for � 1 � t � �: (5.79)We present a solution of this problem with the additional restriction that thedegree of f(t) is bounded by some function h(�): For special cases of �; this solutiongives rise to the known solution [96] of the following problem: for any integer �;1 � � < N; �nd the maximum of U (f) over the class of polynomials f(t) 2 F [t] ofdegree at most � such that f0(U) > 0 andf(t) � 0 for � 1 � t � 1: (5.80)While describing the results, the following polynomialsf (�)2k�1(t) = (t� �)�T 1;0k�1(t; �)�2 (5.81)and f (�)2k (t) = (t� �)(t+ 1)�T 1;1k�1(t; �)�2 (5.82)play the key role. The polynomial f (�)(t) is chosen as one of (5.81) or (5.82) for theproper choice of k. ConsiderM2k�1(�) = U (f (�)2k�1) for t1;0k�1 � � < tk(1 � k < N1;0; t1;00 = �1) andM2k(�) = U (f (�)2k ) for t1;1k�1 � � < t0;1k89

(1 � k < N1;1; t1;10 = �1): Since by (5.70) and (5.14) f (�)2k�1(t)=f (�)2k�1(1) can berepresented as follows:� Uk(t)Tk�1(�; 1)� Uk(�)Tk�1(t; 1)Uk(1)Tk�1(�; 1)� Uk(�)Tk�1(1; 1)� Pk�1i=0 r1;0i U1;0i (t)U1;0i (�)Pk�1i=0 r1;0i U1;0i (1)U1;0i (�) ;using (5.65) and (5.20) we have1U (f (�)2k�1) = �Uk(�)Uk(1)Tk�1(�; 1)� Uk(�)Tk�1(1; 1)and, hence, M2k�1(�) = 1� U1;0k�1(�)Uk(�) ! k�1Xi=0 ri: (5.83)Thus, M2k�1(�) is positive-valued and increases with � in the half-open intervalbeing considered, by Lemmas 5.30 and 5.32. From de�nitions (5.78), (5.81), (5.82),and (5.61) it follows that U (f (�)2k ) may be obtained from U (f (�)2k�1) by replacingall parameters of the system U by those of U0;1 and by multiplying by 2c0;1. Since2c0;1 = 1� 1=U1(�1) (see (2.74)), this givesM2k(�) = �1� 1U1(�1)� 1� U1;1k�1(�)U0;1k (�) ! k�1Xi=0 r0;1i (5.84)and implies that M2k(�) is positive-valued and increases with � in the half-openinterval. On the other hand, by (5.71) and (5.14) f (�)2k (t)=f (�)2k (1) can be representedas follows:� Tk(t; 1)Tk(�;�1)� Tk(�; 1)Tk(t;�1)Tk(1; 1)Tk(�;�1)� Tk(�; 1)Tk(1;�1)� (t+ 1)Pk�1i=0 r1;1i U1;1i (t)U1;1i (�)2Pk�1i=0 r1;1i U1;1i (1)U1;1i (�) ;and by (5.68), (5.10), and (5.65)t+ 12 U1;1i (t) = (t+ 1)T 1;0i (t;�1)2T 1;0i (1;�1) = U1;0i+1(t)U1;0i (�1)� U1;0i (t)U1;0i+1(�1)U1;0i+1(1)U1;0i (�1)� U1;0i (1)U1;0i+1(�1)= Ti+1(t; 1)Ti(1;�1)� Ti(t; 1)Ti+1(1;�1)Ti+1(1; 1)Ti(1;�1)� Ti(1; 1)Ti+1(1;�1) :Using (5.20) we have 1U (f (�)2k ) = Tk(�;�1)Tk(1; 1)Tk(�;�1)� Tk(�; 1)Tk(1;�1)90

and obtain another expressionM2k(�) = 1� U1;0k (�)U0;1k (�)! kXi=0 ri: (5.85)From (5.83)-(5.85), Corollary 5.35, and the equality 1� U1;0k�1(�)Uk(�) ! k�1Xi=0 ri = 1� U1;0k (�)Uk(�) ! kXi=0 ri (5.86)(see (5.64), (5.23), and (5.24)) it follows thatM2k�1(t1;0k ) =M2k(t1;0k ) =M2k+1(t1;0k ) = kXi=0 ri; (5.87)M2k(t1;1k ) =M2k+1(t1;1k ) =M2k+2(t1;1k )= �1� 1U1(�1)� kXi=0 r0;1i = 1� U1;0k (�1)Uk+1(�1)! kXi=0 ri: (5.88)Herewith, by lemmas on separation of roots of adjacent polynomials the correspondingfunctions are de�ned at the points t1;0k and t1;1k .To de�ne the optimal polynomial f (�)(t) we note that by Lemmas 5.29 and 5.30for any k; 1 � k < N � 1; the following inequalities holdt1;1k�1 < t1;0k < t1;1k where t1;10 = �1: (5.89)In the discrete case when tN;N = 1 and tN;1 = �1 we have N1;1 = N � 2 and t1;1N�2 =tN;N�1: We assume that t1;1N�2 = 1 in the in�nite case. As was remarked before, theinequalities (5.89) mean that the half-open interval [�1; t1;1N�2) is partitioned into half-open intervals [t1;1k�1; t1;0k ) and [t1;0k ; t1;1k ); k = 1; :::; N � 2: Let k(�) = kU (�) = k wheneither � 2 [t1;1k�1; t1;0k ) or � 2 [t1;0k ; t1;1k ); and let "(�) = "U (�) = 0 if � 2 [t1;1k�1; t1;0k ) forsome k and "(�) = "U (�) = 1 if � 2 [t1;0k ; t1;1k ) for some k. Any �, �1 � � < t1;1N�2;belongs to the unique half-open interval,t1;1�"(�)k(�)�1+"(�) � � < t1;"(�)k(�) , (5.90)with the number h(�) = 2k(�) + "(�)� 1: (5.91)We also consider the increasing function (d) = U (d) which, in a certain context, isthe inverse to h(�) and maps an integer d (d � 2) to the left end of the half-openinterval with the number d� 1; i.e., (d) = t1;�l�� if d� 1 = 2l� � (5.92)91

where l is an integer and � 2 f0; 1g . (k(t1;�l��) = l; "(t1;�l��) = 1� �; and henceh(t1;�l��) = 2l � �:) For any �, �1 � � < t1;1N�2; we de�ne the polynomialf (�)(t) = (t� �)(t+ 1)" �T 1;"k�1(t; �)�2 (5.93)where " = "(�) and k = k(�): Thus, given � we choose as f (�)(t) (5.81) when "(�) = 0or (5.82) when "(�) = 1 with k = k(�) in the both cases. Note that the polynomial(5.93) has degree h(�) = 2k(�) + "(�) � 1 and satis�es the property (5.79). For any�, �1 � � < t1;1N�2; we consider the functionLU (�) = (f (�)) = f (�)(1)f (�)0 (U) : (5.94)Since t1;0k�1 � t1;1k�1 < t1;0k < tk and t1;1k�1 < t1;0k < t1;1k < t0;1k for 1 � k < N � 1; weobtain LU (�) = � M2k�1(�) if t1;1k�1 � � < t1;0kM2k(�) if t1;0k � � < t1;1k (5.95)and, hence, LU (�) can be represented in the formLU (�) = �1� 1U1(�1)�" 1� U1;"k�1(�)U0;"k (�) ! k�1Xi=0 r0;"i : (5.96)where " = "(�) and k = k(�): Summarizing the properties of the function LU (�); wehave the following statement.Lemma 5.37. The function LU (�) (see (5.94) and (5.96)) is a positive-valued strictlyincreasing continuous function which takes the following values at the left ends of thehalf-open intervals LU (t1;�l��) = �1� 1U1(�1)�� l��Xi=0 r0;�i (5.97)where l is an integer and � 2 f0; 1g:In particular, LU (�1) = 1� 1U1(�1) : (5.98)Using (5.93) and Corollary 5.36 we get the following.Lemma 5.38. (t� �)f (�)(t) = 8><>: c(t+ 1)2 �U1;1k�1(t)�2 if � = t1;1k�12c(t+ 1)�U1;0k (t)�2 if � = t1;0k (5.99)where, for the corresponding �; c = 14 (1� �)f (�)(1) > 0.92

The polynomials f (�)2k�1(t); f (�)2k (t); and f (�)(t) were introduced and the correspond-ing upper bounds on the size of d-codes in polynomial metric spaces were presented bythe author in [64] (see also [67], [68]). Some extremum properties of the polynomialsf (�)2k�1(t) and f (�)2k (t) were found by Sidelnikov in [105]. These results are summarizedin the following statement [68],[71] which, to a certain extent, is a generalization ofthe Gauss-Jacobi formula.Theorem 5.39. For any �; �1 � � < t1;1N�2, the polynomial(t� �)(t + 1)"T 1;"k�1(t; �) (5.100)with k = k(�) and " = "(�) has k + " simple roots �0; �1; : : : ; �k+"�1 (�0 < �1 <: : : < �k+"�1) where �k+"�1 = � and �0 � �1 with equality holding if and only if" = 1 or " = 0 and � = t1;1k�1: Moreover, for any polynomial f(t) of degree at mosth(�) = 2k � 1 + " the following equality holdsf0(U) = 1Z�1 f(t)d�(t) = (LU (�))�1f(1) + k+"�1Xj=0 �(�)j f(�j) (5.101)where (de�ned below by (5.112) and (5.113)) coe�cients �(�)j ; j = 1; : : : ; k + " � 1,are positive and �(�)0 � 0 with equality holding if and only if � = t1;0k :Proof. Using Lemmas 5.29 and 5.30, and the de�nition of k = k(�) and " = "(�),we have t1;"k�1 < t1;1�"k�1+" � � < t1;"k : (5.102)Moreover, in accord with Remark 5.27 and Lemma 5.33,t1;1+"k�1 � t1;1�"k�1+": (5.103)Hence by Corollary 5.7, U1;"k�1(�) > 0 (5.104)and U1;1+"k�1 (�) = T 1;"k�1(�;�1)T 1;"k�1(1;�1) � 0 (5.105)It is easy to see that equality in (5.105) holds if and only if it takes place in (5.102)and (5.103), i.e., when " = 0 and � = t1;1k�1. Since by Theorem 5.4 (for the systemU1;"), (t� �)T 1;"k�1(t; �) = r1;"k�1b1;"k�1 �U1;"k (t)U1;"k�1(�) � U1;"k�1(t)U1;"k (�)� ; (5.106)93

the equation (t� �)T 1;"k�1(t; �) = 0 is equivalent toU1;"k (t)U1;"k�1(t) = U1;"k (�)U1;"k�1(�) :The equality (5.106) for t = �1 and (5.105) show thatU1;"k (�1)U1;"k�1(�1) � U1;"k (�)U1;"k�1(�) (5.107)with equality if and only if " = 0 and � = t1;1k�1. Let t1;"k�1;0 = �1 and t1;"k�1;k =1. Since the ratio U1;"k (t)=U1;"k�1(t) increases in each interval (t1;"k�1;i; t1;"k�1;i+1); i =0; 1; :::; k � 1; from -1 to 1; the polynomial (t � �)T 1;"k�1(t; �) has a zero (which itis convenient to denote) �i+" in each such interval. Moreover, according to (5.107),�" � �1 with equality if and only if " = 0 and � = t1;1k�1, and we have �k�1+" = �,since t1;1k�1 < �: Denoting �0 = �1 when " = 1 we complete the proof of the theoremconcerning the zeros of (5.100).Consider for the polynomial g(t) = (t�1)(t��)(t+1)"T 1;"k�1(t; �) having k+1+ "simple roots �0; �1; : : : ; �k+" (�1 � �0 < �1 < : : : < �k+"�1 = � < �k+" = 1) theLagrange polynomials lj(g; t); j = 0; 1; :::; k+"; of degree k+" such that lj(g;�i) = �i;j ;0 � i; j � k + " (see (5.15)). Using the last statement of Lemma 5.10 we havelk+"(g; t) = (t+ 1)"(t� �)T 1;"k�1(t; �)2"(1� �)T 1;"k�1(1; �) ; (5.108)li+"(g; t) = (t+ 1)"(t� 1)T 1;"k�1(t; �i+")(�i+" + 1)" (�i+" � 1)T 1;"k�1(�i+"; �i+") ; i = 0; 1; :::; k � 1; (5.109)l0(g; t) = (t� 1)(t� �)T 1;1k�1(t; �)2(1 + �)T 1;1k�1(�1; �) when " = 1 and �0 = �1: (5.110)For any polynomial f(t) of degree at most 2k � 1 + "; the polynomialf(t)� k+"Xj=0 f(�j)lj(g; t) (5.111)equals zero at all (simple) roots of g(t) and hence can be represented in the formg(t)h(t) where h(t) is a polynomial of degree at most k � 2. Using the Christo�el-Darboux formulae (5.10) for (t � �)T 1;"k�1(t; �) and the orthogonality relations (5.61)for the system U1;" as well, we can see that 1R�1 g(t)h(t)d�(t) = 0. This givesf0(U) = k+"Xj=0 �(�)j f(�j) where �(�)j = 1Z�1 lj(g; t)d�(t):94

Using (5.70), (5.71), (5.83), (5.85), Theorem 5.4, and (5.61) we �nd that�(�)k+" = (LU (�))�1 ;�(�)i+" = 1c1;" (1 + �i+")" (1� �i+")T 1;"k�1(�i+"; �i+") > 0; i = 0; :::; k � 1; (5.112)�(�)0 = Tk(�; 1)Tk(�1;�1)Tk(�; 1)� Tk(�1; 1)Tk(�;�1) for " = 1 : (5.113)Since U1;0k (�) � 0 with equality if and only if � = t1;0k ; it remains to prove that for� > t1;0k ; Tk(�1;�1)Tk(1; 1)(Tk(1;�1))2 � U0;1k (�)U1;0k (�) > 0and, hence, �(�)0 > 0. This is true because by Lemma 5.31 U0;1k (�)=U1;0k (�) in-creases for t1;0k < � � 1 from �1 to 1 and by the Cauchy inequality (Tk(1;�1))2 �Tk(�1;�1)Tk(1; 1).Corollary 5.40. For any �; �1 � � < t1;1N�2,LU (�) = minU (f) (5.114)where the minimum is taken over the class of polynomials f(t) 2 F [t] of degree atmost h(�) such that f0(U) > 0 and condition (5.79) is ful�lled. Equality in (5.114)takes place if and only if f(t) is proportional to f (�)(t) or to f (�)(t)=(t + 1) when� = t1;�l�� for some integer l and � 2 f0; 1g:Proof. By Theorem 5.39 for any polynomial f(t) of degree at most h(�) = 2k(�) +"(�) � 1 satisfying conditions (5.79) and f0(U) > 0;LU (�) � U (f) (5.115)with equality if and only if all roots of (5.100) are roots of f(t) except the root�0 = �1 when �(�)0 = 0 and, hence, � = t1;0k : However, condition (5.79) impliesthat all roots �1; : : : ; �k+"�2 and �0 when �0 6= �1 must be double. It follows thatequality in (5.115) is attained only for f (�)(t) and for polynomials obtained fromf (�)(t) removing the second root �0 = �1 for � = t1;1k�1 and the same root for � = t1;0k(see (5.93) and (5.99)).Corollary 5.41 ([96]). For any � = 2l� � where l 2 f1; :::; N � 1g and � 2 f0; 1g ;LU (t1;�l��) = �1� 1U1(�1)�� l��Xi=0 r0;�i = maxU (f) (5.116)where the maximum is taken over the class of polynomials f(t) 2 F [t] of degree atmost � such that f0(U) > 0 and condition (5.80) is ful�lled. The maximum in (5.116)takes place if and only if f(t) is proportional to (t+ 1)� �U1;�l��(t)�2 :95

Proof. Let � = t1;�l�� and, hence, k(�) = l, "(�) = 1� �; and h(�) = 2k(�)� "(�) + 1= 2l�� = � . By Theorem 5.39 for any polynomial f(t) of degree at most � satisfyingthe conditions (5.80) and f0(U) > 0;LU (t1;�l��) � U (f) (5.117)with equality if and only if all roots of (5.100) (with � = t1;�l��) are roots of f(t)except for the root �0 = �1 when �(�)0 = 0 and, hence, � = t1;0l : However, in thiscase condition (5.80) implies that all roots �1; : : : ; �k+"�2 and �k+"�1 = � must bedouble and �0 = �1 must be of multiplicity at least � = 1 � "(�). This means thatthe polynomial f(t) of degree at most � = h(�) for which (5.80) is satis�ed must beproportional to t��t+1 f (�)(t) and completes the proof in accord with Lemma 5.38.Dunkl [38] was the �rst who applied this result to design theory.For some applications of Theorem 5.39 and Corollaries 5.40, 5.41 the optimalpolynomial f (�)(t) must have the additional property that all coe�cients f (�)i (U);i = 0; 1; :::; h(�); of its expansion over the system U (not only f (�)0 (U)) are positive.Now we present a certain su�cient condition when this is true.Similar to (5.77) for any polynomial f(t) 2 F [t] we de�ne coe�cientsfi(Ua;b) = ra;bi 1Z�1 f(t)Ua;bi (t)ca;b(1� t)a(1 + t)bd�(t); 0 � i < Na;b: (5.118)Denote by F�(Ua;b) (respectively, F>(Ua;b)) the set of polynomials f(t) 2 F [t] suchthat fi(Ua;b) � 0; 0 � i < Na;b(respectively, such thatfi(Ua;b) > 0; i = 0; 1; :::; h� 1; and fi(Ua;b) = 0; h � i < Na;bfor some h; 1 � h < Na;b). In particular,U1;0i (t) 2 F>(U) (5.119)by (5.65) and (5.14). We say that the system U = fUi(t); 0 � i < Ng satis�es theKrein condition if Ui(t)Uj(t) 2 F�(U) for any i; j; 0 � i; j < N: (5.120)It is clear that if U satis�es the Krein condition, thenf(t)g(t) 2 F>(U) when f(t) 2 F>(U) and g(t) 2 F>(U): (5.121)We say that the system U = fUi(t); 0 � i < Ng satis�es the strengthened Kreincondition if together with (5.120) the following holds:(t+ 1)U1;1i (t)U1;1j (t) 2 F>(U) for any i; j; 0 � i; j < N1;1: (5.122)96

We know (see Corollary 3.13) that in the case of polynomial spaces the system Qconstructed in Section 2.2 satis�es the Krein condition. Moreover, by Lemmas 3.22and 3.25 it satis�es the strengthened Krein condition for in�nite distance-transitivepolynomial spaces (in particular, for the Euclidean sphere Sn�1) and for �nite decom-posable polynomial spaces (in particular, for the Hamming space Hnv and Johnsonspace Jnw ). This also holds [68] for all antipodal spaces de�ned as polynomial spacesX with a standard substitution �(d) for which for every x 2 X there exists a pointx 2 X such that �(d(x; y)) + �(d(x; y)) = 0 for any y 2 X: For antipodal spaces(which include Sn�1, Hnv for v = 2 and Jnw for even n and w = n=2), the system Qhas the following property: Qi(�t) = (�1)iQi(t); i = 0; :::; N � 1:Theorem 5.42. If a system U = fUi(t); 0 � i < Ng satis�es the strengthened Kreincondition, then for any �; �1 � � < t1;1N�2,f (�)(t) 2 F>(U): (5.123)Proof. By Theorem 5.4 and (5.14) (for the system U1;") the polynomial f (�)(t)de�ned by (5.93) can be represented in the form(t+ 1)"r1;"k�1b1;"k�1 �U1;"k (t)U1;"k�1(�)� U1;"k�1(t)U1;"k (�)� k�1Xi=0 r1;"i U1;"i (t)U1;"i (�) (5.124)where " = "(�) and k = k(�): Since t1;0k�1 < t1;1k�1 < t1;0k ; according to Corollary5.7 (for the system U1;0) we have U1;0k (�) < 0 and U1;0i (�) > 0 for i = 0; :::; k � 1when t1;1k�1 � � < t1;0k : Using the Krein condition, (5.119), and (5.121) we obtain thatthe polynomial (5.124) belongs to F>(U) when " = 0 (note that simultaneously weproved that f (�)2k�1(t) 2 F>(U) when t1;0k�1 � � < t1;0k ). For " = 1 and t1;0k � � < t1;1kanalogously we have U1;1k (�) < 0 and U1;1i (�) > 0 for i = 0; :::; k� 1 by Corollary 5.7(for the system U1;1) and then we use the strengthened Krein condition.From the proof it follows that (5.123) is valid if the systems U and U1;1 satisfythe Krein condition. This was noted in [15]. However, (3.89), (3.91), and (3.93) showthat this condition is stronger than the strengthened Krein condition.Theorem 5.43. If a system U = fUi(t); 0 � i < Ng satis�es the Krein condition,then f (�)2k�1(t) 2 F>(U) when t1;0k�1 � � < t1;0k and f (�)2k (t) 2 F>(U) when � = t1;0k .Proof. The �rst statement follows from the last proof. For � = t1;0k by virtue of(5.99), (5.119), and (5.121) it is su�cient to prove that (t + 1)U1;0k (t)=(t � �) canbe expanded over the system U with positive coe�cients. By (5.22) this polynomialcoincides with 2(1� �)Tk(1; 1) �Tk(t; 1)� Tk(t; �) Tk(�1; 1)Tk(�1; �)�97

because they have the same roots and are equal at t = 1: Therefore it remains toprove that �i = 1� Ui(�) Tk(�1; 1)Tk(�1; �) > 0 for i = 0; 1; :::; k:Since t0;1k;k�1 < t1;0k < t0;1k ; �Tk(�1; �)=Tk(�1; 1) = �U0;1k (�) > 0: In general, we donot know the sign of Ui(�); however, for any i = 0; 1; :::; k� 1; Ti(1; �) > 0 and henceUi(�) > Ui+1(�); and �i > �k: Using (5.24) and (5.25) we have�k = Rk(�;�1; 1)Tk(�1; �) = (1� �)Rk(�1; �; 1)2Tk(�1; �) = 1� �2 :Corollary 5.44. Given that the system U satis�es the Krein condition, for any l 2f1; :::; N � 1g and � 2 f0; 1g ;LU (t1;�l��) = �1� 1U1(�1)�� l��Xi=0 r0;�i = minU (f) (5.125)where the minimum is taken over the class of polynomials f(t) 2 F>(U) of degree atmost 2l� � such that condition (5.79) is ful�lled for � = t1;�l��.Corollary 5.45. For any �; �1 � � < t1;0N�2, the polynomialef (�)(t) = (t� �)�T 1;0m�1(t; �)�2 (5.126)where m = k(�) + "(�) or, equivalently, t1;0m�1 � � < t1;0m (t1;00 = �1); has properties(5.79), ef (�)0 > 0; and also ef (�)(t) 2 F>(U) if the system U satis�es the Krein condition.The function eLU (�) = 1� U1;0m�1(�)Um(�) !m�1Xi=0 ri with m = k(�) + "(�)being equal to U � ef (�)�is a positive-valued strictly increasing continuous functionwhich takes the following values at the left ends of the half-open intervals:eLU (t1;�l��) = LU (t1;�l��) = �1� 1U1(�1)�� l��Xi=0 r0;�i (5.127)where l is an integer and � 2 f0; 1g: 98

Thus, eLU (�) = LU (�) if t1;1k�1 � � � t1;0k and it is possible to show [67] thateLU (�) > LU (�) if t1;0k < � < t1;1k (this is not a consequence of Corollary 5.40 becausethe polynomial ef (�)(t) has degree 2k+1 = h(�)+1 when � lies inside the last interval).One more extremum property of polynomials f (�)2k�1(t) and f (�)2k (t) was found bySidelnikov [105] (see also [68]).Corollary 5.46. For any � = 2l� 1 + � with an integer l and � 2 f0; 1g;t0;�l = sup�where the supremum is taken over all �; �1 � � < 1; for which there exists apolynomial f(t) of degree at most � such that f0(U) > 0 and f(t) � 0 when �1 �t � �:This result was used in [39] to prove that the inequality of Theorem 4.14, which ex-tends Tiet�av�ainen's result [117] for the Hamming space to the case of any polynomialspace (see Remark 4.16), cannot be improved by the method considered.Note that by Corollary 5.40 f (�)(t) also minimizes U (f) = f(1)=f0 over theclass of polynomials f(t) = h(�)Pi=0 fiUi(t) such that f0 > 0; fi � 0; i = 1; :::; h(�); andcondition (5.79) is ful�lled. Using Theorem 5.39 Boyvalenkov, Danev, and Bumova[20] (see also [21]) obtained necessary and su�cient conditions for the optimality off (�)(t) over the same class of polynomials but without the restriction that their degreedoes not exceed h(�). To describe this result, for any �; �1 � � < t1;1N�2, we consideron F [t] the following linear functional G� :G�(f) = (LU (�))�1f(1) + k(�)+"(�)�1Xj=0 �(�)j f(�j):By Theorem 5.39, G�(f) = f0(U) for any polynomial f(t) of degree at most h(�) andG�(f) = (LU (�))�1f(1) if f(t) is divisible by f (�)(t).Theorem 5.47 ([20]). Provided f (�)(t) =Ph(�)j=0 f (�)i Ui(t) 2 F>(U) (Theorem 5.42gives a su�cient condition for this property),LU (�) = minU (f);where the minimum is taken over the class of polynomials f(t) 2 F [t] such that(i) f0(U) > 0; fi(U) � 0; i = 1; :::; deg f;(ii) f(t) � 0 for �1 � t � �;if and only if G�(Uj) � 0 for all j; h(�) < j < N:99

Proof. Let the condition of the theorem be ful�lled, where f(t) =Pli=0 fiUi(t) hasproperties (i) and (ii). By Corollary 5.40 we can assume that l > h(�): Applying G�to Ph(�)i=0 fiUi(t) we getf0 = (LU (�))�1f(1) + h(�)Xj=0 �(�)j f(�j)� lXj=h(�)+1 fjG�(Uj) � (LU (�))�1f(1)and, hence, (f) � LU (�): Let, conversely, G�(Uj) < 0 for some j; h(�) < j < N:There exist unique polynomials a(t) of degree j � h(�) and b(t) = Ph(�)�1i=0 biUi(t)such that Uj(t) = f (�)(t)a(t)+ b(t): Consider f(t) = f (�)(t)(a(t)+ c) = Uj(t)� b(t)+cf (�)(t), where c = max � min�1�t�� a(t); max0�i�h(�)�1 bif (�)i ; 0! :This choice of c implies that fi(U) � 0; i = 0; :::; j; and f(1) > 0 (we used f (�)i > 0;i = 0; :::; h(�)) and that f(t) � 0 for �1 � t � � because a(t)+c � 0 and f (�)(t) � 0in this range. Since G�(f) = (LU (�))�1f(1) = G�(Uj)+f0(U) < f0(U) and f(1) > 0;we have f0(U) > 0 and (f) < LU (�). This completes the proof.5.5. Applications to polynomial metric spaces and polynomial graphsFor a compact metric space X = (X; d(x; y)) we denoted by A(X; d) the maximumsize of a code C � X with minimal distance d(C) � d (see (2.1)). Considering on X ameasure � and �xing a continuous strictly monotone real function �(d) (substitutionfunction), by means of (2.12) we de�ned a weighted � -design C = (C;m) where m isa certain positive-valued function on C; Px2Cm(x) = jCj; and, in particular, a (simple)� -design C = (C;m) for which m(x) � 1 for all x 2 C (see also (3.58) for polynomialspaces). The value d0(C) = �(C) + 1 was called the dual distance of C where �(C)is the strength of C (see (2.13)). We denoted by B(X; d) the minimum size of a(weighted) set C � X with dual distance d0(C) � d (see (2.14)). In Section 2.2we assumed that in the in�nite case the mean measure �(d) of closed metric balls ofradius d and the substitution function �(d) have continuous derivatives �0(d) and �0(d)which are not zero when 0 < d < D(X) (D(X) is the diameter of X). Furthermore,we remarked that without loss of generality it is possible to assume that �(d) isthe standard substitution, i.e., a continuous strictly decreasing function on [0; D(X)]such that �(D(X)) = �1 � �(d) � �(0) = 1: (The de�nition of a weighted � -designand the property of a metric space to be polynomial are invariant under any lineartransformation of �(d).) Using these functions �(d) and �(d), and Theorem 2.7 weconstructed on the interval [�1; 1] a system Q of orthogonal polynomials Qi(t) of100

degree i and a system of positive constants ri (i = 0; 1; :::; s(X)) such thatri 1Z�1 Qi(t)Qj(t)d�(t) = �i;j ; (5.128)Qi(1) = 1; i = 0; 1; :::; s(X);where �(t) = 1��(��1(t)) and r0 = 1. The system Q satis�es all conditions imposedon the system U considered in Sections 5.1-5.4. In particular, in the in�nite case(5.128) takes the form ri 1Z�1 Qi(t)Qj(t)w(t)dt = �i;j ;where w(t) = ��0(d)=�0(d) and d = ��1(t), and in the �nite case when �(X) =fd0; d1; ::: ; dsg, d0 = 0 < d1 < ::: < ds = D(X); s = s(X); �(t) has N = s+ 1 stepsat the points tN;l = �(dN�l) with sizes wl = 1jXj2 Px2XBdN�l(x;X) > 0; NPl=1wl = 1;and (5.128) takes the formri NXl=1 Qi(tN;l)Qj(tN;l)wl = �i;j :(Note that in the latter case QN (t) = (t� �(d0)):::(t� �(ds)); N0;1 = N1;0 = N � 1;N1;1 = N � 2; and t1;1N�2 = �(d1).) We proved that for polynomial metric spaces thesystem Q satis�es the Krein condition and for in�nite distance transitive spaces anddecomposable graphs it satis�es the strengthened Krein condition (Lemmas 3.22 and3.25). Using the fact that for any polynomial f(t) and for any (weighted) set C � Xthe equality Xx;y2C f(�(d(x; y)))m(x)m(y) = jCj s(X)Xi=0 fi(Q)ri B0i(C) (5.129)holds and all coe�cients B0i(C) are nonnegative for polynomial spaces, we provedTheorems 3.14 and 3.15. According to these theorems, for any code C with minimaldistance d(C) � d in a polynomial space X and for any polynomial f(t) such thatf0(Q) > 0; fi(Q) � 0 for i = 1; :::; s(X) and f(t) � 0 for �1 � t � �(d); the followingholds jCj � Q(f) = f(1)f0(Q) (5.130)with equality if and only if the polynomial f(t) is annihilating for C andfi(Q)B0i(C) = 0 for i = 1; :::; s(X): (5.131)101

Analogously, for any weighted � -design C = (C;m) in a polynomial space X andfor any polynomial f(t) such that f0(Q) > 0; fi(Q) � 0 for i = � + 1; :::; s(X) andf(t) � 0 for �1 � t � 1; the following holdsjCj � Q(f) = f(1)f0(Q) (5.132)with equality if and only if C is a simple � -design, the polynomial f(t) is annihilatingfor C and fi(Q)B0i(C) = 0 for i = � + 1; :::; s(X): (5.133)This reduces the problem of obtaining universal bounds on A(X; d) and B(X; d) bythe method considered to the �-packing and � -design problems for the system Q ofwhich exploratory discussion is contained in Section 3.3. Considering the system Q(and the system P for polynomial graphs) as U we obtain the corresponding boundsas corollaries of the results on orthogonal polynomials given above. A code C ina polynomial space X is referred to as a maximal code if A(X; d(C)) = jCj and aminimal design if B(X; d0(C)) = jCj:Theorem 5.48. Let X be a polynomial space with standard substitution �(d) andsuppose the system Q de�ned by Theorem 2.7 satis�es the strengthened Krein con-dition. Then for any code C � X (with d(C) > d1 in the �nite case) the followingholds jCj � LQ(�(d(C))) (5.134)where by Lemma 5.37 LQ(�) is a positive-valued strictly increasing continuous func-tion de�ned by (5.96). The bound (5.134) is attained if and only if �(C) = 2k� 1+ "and fC(t) = t� �1� � � t+ 12 �"� T 1;"k�1(t; �)T 1;"k�1(1; �) (5.135)where � = �(d(C)); � = �(C); k = k(�); " = "(�):Proof. By Corollary 5.40 and Theorem 5.42 the polynomial f (�)(t) = (t � �)(t +1)" �T 1;"k�1(t; �)�2 of degree h(�) = 2k � 1 + " has the desired properties and (5.130)implies (5.134) with equality if and only if (t��)(t+1)"T 1;"k�1(t; �) is annihilating forC and (5.131) holds. This proves (5.134) and the su�ciency of the stated conditions.Now we prove their necessity. Note that �(C) < s(X) since by Corollary 3.20 �(C) =s(X) implies that C = X and d(C) = d1: Since f (�)(t) 2 F>(Q) and �(C) < s(X),the condition (5.131) means that h(�) < s(X) and �(C) � h(�): Therefore, usingLemma 2.17 we have 2k � 1 + " � �(C) � 2s(C)� �and hence s(C) > k�1+"�:On the other hand, the polynomial (5.135) of degree k+"�is also annihilating for C because �1 is not a root of the minimal polynomial for C if102

� = 0: This means that s(C) = k+"� and (5.135) is minimal for C: By the de�nitionsof k = k(�) and " = "(�); � < t1;"k and hence by Lemma 5.37, jCj = LQ(�) < LQ(t1;"k ):In the case �(C) � 2k+ " by Theorem 2.16 and (5.97) we would have jCj � LQ(t1;"k ):Hence �(C) = 2k � 1 + ":The polynomial (5.135) can have only simple roots �0; �1; : : : ; �k+"�1 (�1 � �0 <�1 < : : : < �k+"�1 = �) and hence only di; i = 1; :::; k+ "; de�ned by �(di) = �k+"�ican be nonzero distances of C. Therefore, (3.66) holds if we replace fC(t) by thepolynomial g(t) de�ned in Theorem 5.39 whose degree does not exceed �(C) + 1.Applying (5.101) to all the Lagrange polynomials (5.109)-(5.110) we can describe thedistance distribution of any code C; for which (5.134) is attained, in terms of thecoe�cients �(�)j of Theorem 5.39.Corollary 5.49. For any code C for which (5.134) is attained,Bdi(C) = LQ(�) � �(�)k+"�i ; i = 1; :::; k + "; (5.136)where � = �(d(C)); k = k(�); " = "(�); and the coe�cients �(�)i are the same as inTheorem 5.39.Note that by Theorem 5.39 all �(�)i are positive except the cases � = t1;0k when�(�)0 = 0. Since by (5.135) s(C) = k(�) + "(�)�(C); then the condition �(C) =2k(�)� 1 + "(�) can be rewritten in the form�(C) = 2s(C)� �(C)� �"(�);�(C) (5.137)and gives rise to tight designs if (and only if) "(�) 6= �(C): We consider these signif-icant special cases of the universal bound (5.134) and verify that they are valid forany polynomial space. We use the notation da;bi = ��1(ta;bi ) (see (2.71)).Theorem 5.50. For any code C in a polynomial space X such that d(C) � d1;�l�� forsome l 2 f1; :::; s(X)g and � 2 f0; 1g;jCj � LQ(t1;�l��) = �1� 1Q1(�1)�� l��Xi=0 r0;�i (5.138)with equality if and only if�(C) = 2s(C)� �(C) and fC(t) = �1 + t2 ��Q1;�l��(t): (5.139)Proof. If � = t1;�l��; then k(�) = l; "(�) = 1��; and h(�) = 2l��: Taking into accountthat for any polynomial space f (�)2l��(t) 2 F>(Q) when � = t1;�l�� by Theorem 5.43, wecan repeat the proof of Theorem 5.48 for this case and use Corollary 5.44 to obtain103

(5.138). If conditions (5.139) are satis�ed, then d(C) = d1;�l��; s(C) = l; �(C) = �;�(C) = 2l��; and by Theorem 2.16 we have equality in (5.138). If equality in (5.138)holds, then �(C) = 2k(�) � 1 + "(�) = 2l � � and the same theorem proves thenecessity of conditions (5.139).Corollary 5.51. Any tight � -design C in a polynomial space X is a maximal code.The optimality of the choice f (�)(t); for the method based on using the inequality(5.130), can be expressed in the form of the following statement which is a directconsequence of Corollaries 5.40 and 5.44.Corollary 5.52. For any �; �1 � � < t1;1N�2, and any polynomial f(t) 2 F [t] ofdegree at most h(�) such that f0(U) > 0 and f(t) � 0 for �1 � t � �(d) which isnot proportional to f (�)(t) or to f (�)(t)=(t+1) when � = t1;�l�� for some integer l and� 2 f0; 1g; the following holds:Q(f) = f(1)f0(Q) > LQ(�):Since the question of whether the system Q corresponding to a polynomial met-ric space satis�es the strengthened Krein condition is still open (1997), we can alsoconsider for any �, �1 � � < t1;0s�1; the polynomialef (�)(t) = (t� �)�T 1;0m�1(t; �)�2 (5.140)of odd degree 2m� 1 where m = k(�) + "(�) or, equivalently, t1;0m�1 � � < t1;0m . ByTheorem 5.43 and Corollary 5.45 this polynomial has the properties ef (�)(t) 2 F>(Q)and ef (�)(t) � 0 if �1 � t � �; and the functioneLQ(�) = Q( ef (�)) = 1� Q1;0m�1(�)Qm(�) !m�1Xi=0 ri:is a positive-valued strictly increasing continuous function. Herewith,eLQ(�) = LQ(�) if "(�) = 0 (5.141)and eLQ(t1;�l��) = LQ(t1;�l��) = �1� 1Q1(�1)�� l��Xi=0 r0;�i (5.142)where l is an integer and � 2 f0; 1g:Corollary 5.53. For any code C in a polynomial space,jCj � eLQ(�) = 1� Q1;0m�1(�)Qm(�) !m�1Xi=0 ri: (5.143)where � = �(d(C)) and m = k(�) + "(�):104

According to Lemma 5.38 for any l 2 f1; :::; s(X)g and � 2 f0; 1g the polynomialf(t) = t� �t+ 1 f (�)(t) where � = t1;�l�� (5.144)has degree h(�) = 2l � � and up to a constant multiple is equal to the polynomial(t+1)� �U1;�l��(t)�2and hence to the polynomial g(2l��)(t) de�ned by (2.72). Moreover,by Theorem 5.39, (f) = (f (�)) = LQ(t1;�l��):Thus the polynomial (5.144) gives rise to the same solution of the restricted (2l� �)-design problem as in Theorem 2.16. However, Corollary 5.41 also �nds the optimalityand the uniqueness of the solution.Corollary 5.54. For any (2l � �)-design C in a polynomial space X where l 2f1; :::; s(X)g and � 2 f0; 1g ;jCj � LQ(t1;�l��) = �1� 1Q1(�1)�� l��Xi=0 r0;�i (5.145)with equality in (5.145) if and only if fC(t) =( t+12 )� Q1;�l��(t): For any polynomialf(t) 2 F [t] of degree at most � = 2l � � such that f0(Q) > 0 and f(t) � 0 for�1 � t � 1 which is not proportional to (5.144), the following holds:Q(f) = f(1)f0(Q) < LQ(t1;�l��):Using the function (d) (see (5.92)) we can express (5.145) as followsjCj � LQ( Q(d)) if d0(C) � d: (5.146)Furthermore, we can rewrite (5.138) in the formjCj � LQ( Q(d)) if d(C) � ��1Q ( Q(d)): (5.147)We know that for any code C in a compact metric space X;�(C) � 2s(C)� �(C) (5.148)and equality in (5.148) is necessary for C to be a tight design. For polynomial spacesthe equality in (5.148) is already su�cient for attainability of the bounds (5.138) and(5.145), and the equality (5.137) is su�cient for attainability of the bound (5.134).Theorem 5.55 ([68]). Let C be a code in a polynomial space X , � = �(C) � 1;s = s(C) � 1; � = �(C); � = �Q(d(C)); k = kQ(�); " = "Q(�); and hencet1;1�"k�1+" � � < t1;"k :105

Then � = 2s� � if and only if s = k; � = 1� " and� = t1;1�"k�1+"; jCj = LQ(�); fC(t) = ( t+ 12 )�Q1;�k��(t) (5.149)and � = 2s� 1� � if and only if s = k + "; � = " and� 6= t1;1�"k�1+"; jCj = LQ(�); fC(t) = t� �1� � � t+ 12 �� T 1;�k�1(t; �)T 1;�k�1(1; �) : (5.150)In the both cases fC(t) 2 F>(Q):The proof in the case � = 2s follows from the bound for designs (Theorem 2.16)and the absolute bound of Delsarte (Theorem 3.18). A similar proof would be validin the case � = 2s � � with � = 1 if Conjecture 3.26 is true. In this case and inthe general case Theorems 2.16, 3.17, 3.18, Corollary 5.53 monotonicity of LQ(�)and properties of adjacent systems of orthogonal polynomials only were used. (Thestrengthened Krein condition is not needed.)Theorem 5.55 describes some properties of the class of codes C in a polynomialspace X de�ned by the condition �(C) � 2s(C) � 1 � �(C) � 1 (cf. (5.148)). ByTheorem 3.21 this class consists of distance-regular codes. By Theorem 5.48 any codeC; for which (5.134) is attained, belongs to this class. On the other hand, Theorems5.48, 5.55 and (5.141)-(5.143) imply the following results on the optimality of codesof this class.Corollary 5.56. In a polynomial space X; any code C such that �(C) � 2s(C) � 1is maximal.Corollary 5.57. In a polynomial space X with the system Q satisfying the strength-ened Krein condition, any code C such that �(C) � 2s(C) � 2 and �(C) = 1 ismaximal.Remark 5.58. It should be noted that the main parameters of a hypothetical codeC, for which the inequality (5.134) is attained (i.e., jCj = LQ(d(C))), can be uniquelyde�ned by means of only the parameter d = d(C) and some parameters of the systemQ for the whole space X . Indeed, let � = �Q(d); k = kQ(�); " = "Q(�): Thenby Theorems 5.48 and 5.55 s(C) = k; �(C) = 1 � " and �(C) = 2s(C)� �(C) if� = t1;1�"k�1+"; and s(C) = k + "; �(C) = " and �(C) = 2s(C)� �(C)� 1 if � 6= t1;1�"k�1+".In the both cases the set �0(C) = fd1; :::; dsg of s = s(C) distances between di�erentcode points and the distance distribution of C can be found (see Corollary 5.49).Finally, C is a distance-regular code and its intersection numbers (2.9) are given by(3.67) and (3.68). Since the intersection numbers (and also the distances di in the caseof polynomial graphs) are integers, these arguments can be used to prove nonexistenceof a code C, for which (5.134) is attained. For example, it was observed [22] that inthe case of antipodal spaces and "Q(�) = 1; by Corollary 5.49 Bdk+1(C) < 1: Hence,106

under these conditions, the inequality (5.134) might be attained only for tight 2k-designs when � = t1;0k�1 and Bdk+1(C) = 0. On the other hand, this approach can beconsidered as a general part of a proof of the uniqueness (up to isometry) of codeswith di�erent minimal distances and in di�erent polynomial spaces for which (5.134)is attained (see, for example, [78, Chapt. 20], [28, Chapt. 14]).Now we consider a polynomial graph X with standard substitutions �Q(d) and�P (d) (see (4.11)) and formulate statements which follow from the results given aboveand the duality in bounding the optimal sizes of codes and designs (Section 4.3). Weshall write kU (�); "U (�); LU (�); U (d); ta;bk (U) = �U (da;bk (U)) and T a;bk (t; �; U) forU = Q and U = P to explain for what system these values are considered.Theorem 5.59. For any code C in a polynomial graph X;LQ( Q(d0(C))) � jCj � jX jLP ( P (d(C))) (5.151)with equality in the left-hand side if and only ifd0(C) = 2s(C)� �(C) + 1 and fC(t) = �1 + t2 ��(C)Q1;�(C)s(C)��(C)(t) (5.152)and with equality in the right-left side if and only ifd(C) = 2s0(C)� �0(C) + 1 and efC(t) = �1 + t2 ��0(C) P 1;�0(C)s0(C)��0(C)(t): (5.153)Codes for which the upper bound in (5.151) is attained are called perfect codes.Thus we consider perfect codes with odd and even minimal distance (similar to tightdesigns de�ned as codes for which the lower bound in (5.151) is attained).Theorem 5.60. For any code C in a polynomial graph X such that d(C) � d1;�l��(Q)and d0(C) � d1;#m�#(P ) for some l;m 2 f1; :::; s(X)g and �; # 2 f0; 1g;jX jLP ( P (2m� #+ 1)) � jCj � LQ( Q(2l � � + 1)) (5.154)with equality in the right-hand side if and only ifd0(C) = 2s(C)� �(C) + 1 and fC(t) = �1 + t2 �� Q1;�l��(t) (5.155)and with equality in the left-hand side if and only ifd(C) = 2s0(C) � �0(C) + 1 and efC(t) = �1 + t2 �# P 1;#m�#(t): (5.156)107

Corollary 5.61. In a polynomial graph X any perfect code is a minimal design andany tight design is a maximal code.Theorem 5.62. Let X be a polynomial graph for which the systems Q and P satisfythe strengthened Krein condition. Then for any code C � X with d(C) > 1 andd0(C) > 1 the following holds:jX jLP (�P (d0(C))) � jCj � LQ(�Q(d(C))) (5.157)with equality in the right-hand side if and only ifd0(C) = 2k + " and fC(t) = t� �1� � � t+ 12 �"� T 1;"k�1(t; �;Q)T 1;"k�1(1; �;Q) (5.158)where � = �Q(d(C)); � = �(C); k = kQ(�); " = "Q(�); and with equality in theleft-hand side if and only ifd(C) = 2k + " and efC(t) = t� �1� � � t+ 12 �"�0 T 1;"k�1(t; �; P )T 1;"k�1(1; �; P ) (5.159)where � = �P (d0(C)); �0 = �0(C); k = kP (�); " = "P (�):For polynomial graphs for which Q and/or P do not satisfy the strengthened Kreincondition one can use Corollary 5.53 and the corresponding dual analog. Similarto the case of polynomial spaces we can use Theorems 5.59, 4.13, Corollary 4.19,monotonicity of LP (�) and properties of adjacent systems of orthogonal polynomialsto obtain the following statements (cf. Theorem 5.55).Theorem 5.63. Let C be a code in a polynomial graph X , d = d(C) � 2; s0 =s0(C) � 1; �0 = �0(C); � = �P (d0(C)); k = kP (�); " = "P (�); and hencet1;1�"k�1+"(P ) � � < t1;"k (P ):Then d = 2s0 � �0 + 1 if and only if s0 = k; �0 = 1� " and� = t1;1�"k�1+"(P ); jCj = jX jLP (�) ; efC(t) = ( t+ 12 )�0P 1;�0k��0(t); (5.160)and d = 2s0 � �0 if and only if s0 = k + "; �0 = " and� 6= t1;1�"k�1+"(P ); jCj = jX jLP (�) ; efC(t) = t� �1� � � t+ 12 ��0 T 1;�0k�1 (t; �; P )T 1;�0k�1 (1; �; P ) : (5.161)In the both cases efC(t) 2 F>(P ): 108

Thus the class of codes C in a polynomial graphX de�ned by the condition d(C) �2s0(C) � �0(C) � 2 (cf. the statement 3 of Theorem 4.5) has the following property:each of the values d0(C) or jCj uniquely determines the other as well as the valuesd(C); s0(C); �0(C); and the dual-annihilating polynomial efC(t) (and hence the set ofs0(C) integers which are dual distances). Note that the condition d(C) � 2s0(C) ��0(C) (as distinguished from the condition d0(C) � 2s(C)� �(C); see Theorem 3.21and Remark 5.58) does not imply that C is a distance-regular code. In particular,any perfect code C � Hn2 for which d(C) = 2s0(C)��0(C) = 3 is not distance-regularfor n > 7 (appears in [6]). We also have the following results on the optimality ofcodes of this class.Corollary 5.64. In a polynomial graph X; any code C such that d(C) � 2s0(C) isa minimal design.Corollary 5.65. In a polynomial graphX with the system P satisfying the strength-ened Krein condition, any code C such that d(C) � 2s0(C) � 1 and �0(C) = 1 is aminimal design.One more pair of universal bounds for designs and codes in polynomial graphs waspresented in Theorem 4.27.6. Summary for the basic polynomial spaces6.1. The unit Euclidean sphere and the projective spacesThe unit Euclidean sphere in Rn (see Examples 2.3, 2.10, 3.11)Sn�1 = (x = (x1; :::; xn) 2 Rn; nXi=1 x2i = 1)with Euclidean distance d(x;y) = pPni=1(xi � yi)2 is a compact metric space ofessential interest in coding theory and information theory. We can also measure thedistance between x;y 2 Sn�1 by the angular distance '(x;y) wherecos'(x;y) = (x;y) = nXi=1 xiyi = 1� 12d2(x;y): (6.1)Denoting the normalized Lebesgue measure on Sn�1 (the normalized surface area) by�; we state that Sn�1 is a distance invariant space and�(d) = �n�1(')�n�1 if cos' = 1� d2=2; (6.2)109

where �n�1(') is the surface area of a spherical cap on Sn�1 of radial angle ' and�n�1 = 2�n�1(�2 ) is the surface area of Sn�1: The function�(d) = 1� d22 (6.3)is the standard substitution for the Euclidean sphere X = Sn�1. Using (2.34), (6.2),(2.20), (6.3) we can see that the corresponding system Q consists of polynomialsorthogonal on the interval [-1,1] with respect to the weight functionw(t) = �0(t) = �(n2 )�(n�12 )�( 12 ) (1� t2)n�32 :As we remarked, the properties (2.49) and (2.50) of the normalized Jacobi polynomialsP�;�i (t) de�ned by (2.48) imply that for the case Sn�1;Qi(t) = P n�32 ;n�32i (t); i = 0; 1; ::: , (6.4)and ri = 2i+n�2i+n�2 �i+n�2i �: (The Jacobi polynomials P�;�i (t) with � = � are calledGegenbauer polynomials.) By Lemma 3.22 the system (6.4) satis�es the strengthenedKrein condition.The fact that Sn�1 is a polynomial space with standard substitution (6.3) is wellknown (see, for example, [14]) and can be proved using the approach described in theend of Section 3.2. The isometry group of Sn�1 is the group O(n) of orthogonal ma-trices of order n and Sn�1 is distance-transitive with respect to O(n): The subspacesVi ; i = 0; 1; ::: , on which the components of representation L(g) de�ned by (3.75)act, consist of all functions on Sn�1 which are represented by harmonic homogeneouspolynomials h(x1; :::; xn) of total degree i in variables x1; :::; xn: For the zonal spher-ical function �i(d) (see (3.76)) there exists a polynomial Ui(t) of degree i such that�i(d(x;y)) = Ui((x;y)). Hence Sn�1 is polynomial with respect to (6.3) and Ui(t)coincides with Qi(t) = P n�32 ;n�32i (t):In the case of X = Sn�1 the concept of a (weighted) � -design C = (C;m) isconnected [35] (see also (3.58)) with the approximation formula for the evaluation ofmulti-dimensional integrals over Sn�1 of the following sortZSn�1 u(x)d�(x) � 1Px2Cm(x) Xx2C u(x)m(x): (6.5)The (weighted) set C = (C;m) is a (weighted) � -design in Sn�1 with respect tothe substitution �(d) = 1 � d2=2 (such �-designs are called spherical) if and only ifthe approximation formula (6.5) becomes equality for all functions u(x) which arepolynomials in coordinates of x = (x1; :::; xn) 2 Sn�1 of total degree at most �: ThusB(Sn�1; � +1) is the minimum number of nodes in the approximation formula of thistype. 110

The relationship (6.1) allows us to consider A(Sn�1; d) as the maximum sizeM(n; ') of a spherical code C � Sn�1 with the minimal angular distance '(C) = 'between its distinct points where cos' = 1 � d2=2 and d = 2 sin '2 : There exists aclose connection between the value M(n; ') and some classical packing problems inRn: We consider the compact (but not distance invariant) metric spaceRn� = (x = (x1; :::; xn) 2 Rn; nXi=1 x2i � �)with the Euclidean distance. For the measure � equal to n-dimensional volume, it isknown (see, for example, [40]) that �(Rn� ) = �n�(Rn1 ): Therefore the ratioA(Rn� ; 2)�(Rn1 )�(Rn� ) = A(Rn� ; 2)�ncharacterizes the density of packing the ball Rn� of radius � by balls of radius 1. Thehighest density of a sphere packing in Rn is de�ned as follows:�n = lim�!1 A(Rn� ; 2)�n : (6.6)This limit really exists and does not change if we replaceA(Rn� ; 2) in (6.6) byA(Rn��1; 2):Another characteristic of the space Rn is the kissing number Mn equal to the maxi-mum number of unit spheres in Rn that touch one such sphere and do not intersect.The following statements are valid:Mn =M(n; �3 ); (6.7)�n�1 = lim'!0 M(n; 2')�n�1(')�n�1 ; (6.8)�n �M(n+ "; 2') sin n' for any '; 0 < ' � �2 ; (6.9)where " = 0 if �6 � ' � �2 ; and " = 1 if ' < �6 ; and, in particular,�n �Mn2�n (6.10)((6.7) is evident and proofs of (6.8)-(6.9) can be found in [67]).Now we apply the results of Section 5.5 to bound M(n; ') = A(Sn�1; 2 sin '2 ) andB(Sn�1; � + 1). Let Qi(t) = Q(n)i (t) = P n�32 ;n�32i (t); i = 0; 1; ::: , and let p�;�i bethe largest root of the Jacobi polynomial P�;�i (t): Thus for the system Q we haveti = pn�32 ;n�32i ; t1;0i = pn�12 ;n�32i ; t1;1i = pn�12 ;n�12i :Using the known properties of the Gegenbauer polynomials we have111

ri = 2i+ n� 2i+ n� 2 �i+ n� 2i �; bi = i+ n� 22i+ n� 2 ; ai = 0and hence (see Theorem 5.3 and (5.9)) the following recurrence(i+ n� 2)Qi+1(t) = (2i+ n� 2)tQi(t)� iQi�1(t): (6.11)The �rst seven polynomials Qi(t) = Q(n)i (t) have the following form:Q0(t) = 1; Q1(t) = t; Q2(t) = nt2 � 1n� 1 ; Q3(t) = (n+ 2)t3 � 3tn� 1 ;Q4(t) = (n+ 2)(n+ 4)t4 � 6(n+ 2)t2 + 3n2 � 1 ;Q5(t) = (n+ 4)(n+ 6)t5 � 10(n+ 4)t3 + 15tn2 � 1 ;Q6(t) = (n+ 4)(n+ 6)(n+ 8)t6 � 15(n+ 4)(n+ 6)t4 + 45(n+ 4)t2 � 15(n+ 3)(n2 � 1) :Using (2.77) and (6.4) we can �nd thatkXi=0 r0;�i = �k + n� 1k �+ (1� �)�k + n� 2k � 1 �; where � 2 f0; 1g :By (5.10) and Lemma 5.24 we can expressQ1;0k (t); Q0;1k (t); and Q1;1k (t) via polynomialsof the system Q as followsQ1;0k (t) = (n� 1) (Qk(t)�Qk+1(t))(2k + n� 1) (1� t) ; Q0;1k (t) = Qk(t) +Qk+1(t)1 + t ;Q1;1k (t) = 2k(Qk�1(t)�Qk+1(t))(2k + n� 2)(1� t2) :In particular, this givest1;10 = �1; t1;11 = 0; t1;12 = 1pn+ 2 ; t1;13 = p3pn+ 4 ;t1;14 =s3(n+ 4) +p6(n+ 3)(n+ 4)(n+ 4)(n+ 6) ; t1;15 =s5(n+ 6) +p10(n+ 3)(n+ 6)(n+ 6)(n+ 8) ;t1;01 = � 1n; t1;02 = pn+ 3� 1n+ 2 :112

These calculations show that the function LQ(�) = L(n)Q (�) for the case of the Euclid-ean sphere Sn�1 can be expressed as follows:L(n)Q (�) = 8<: �k+n�3k�1 � �2k+n�3n�1 � Qk�1(�)�Qk(�)(1��)Qk(�) � if t1;1k�1 � � < t1;0k ;�k+n�2k � �2k+n�1n�1 � (1+�)(Qk(�)�Qk+1(�))(1��)(Qk(�)+Qk+1(�))� if t1;0k � � < t1;1k ; (6.12)and takes the following values at the ends of the half-open intervalsL(n)Q (t1;1k�1) = 2�k + n� 2k � 1 �; L(n)Q (t1;0k ) = �k + n� 1k �+�k + n� 2k � 1 �: (6.13)Using Theorem 2.16, Corollary 5.54 and (3.58) one can extend the known Delsarte-Goethals-Seidel bound [35] to the case of weighted spherical designs.Theorem 6.1 (DGS bound). For any (weighted) (2l� �)-design C in Sn�1 wherel = 1; 2; ::: , and � 2 f0; 1g ;jCj � L(n)Q (t1;�l��) = �l + n� 2n� 1 �+�l + n� 1� �n� 1 � (6.14)with equality in (6.14) if and only if C is a simple design and fC(t) =( t+12 )� Q1;�l��(t):Spherical codes C for which (6.14) is attained are called tight spherical designsand characterized by the condition �(C) = 2s(C)� �(C) according to Theorem 5.55.Theorems 5.48, 5.50 and 5.55 and Lemma 3.22 imply the following statement.Theorem 6.2 ([64], [65], [68]). Let C be a code in Sn�1 with the minimal angulardistance ' = '(C) or with the minimal distance d(C) = 2 sin '2 ; � = �Q(d(C)) =cos'; k = kQ(�); " = "Q(�) and hence t1;1�"k�1+" � � < t1;"k : ThenjCj � L(n)Q (cos') (6.15)where LQ(�) is de�ned by (6.12) and, in particular,jCj � �l+ n� 2n� 1 �+�l + n� 1� �n� 1 � if cos' � t1;�l�� (6.16)for some l = 1; 2; ::: , and � 2 f0; 1g : The bound (6.15) is attained if and only if�(C) � 2s(C)� �(C)� 1 � 1:If jCj = L(n)Q (cos') and cos' = t1;1�"k�1+"; then s(C) = k; �(C) = 1 � "; �(C) =2s(C) � �(C) (and hence C is a tight design), fC(t) =( t+12 )1�" Q1;1�"k�1+"(t); if jCj =L(n)Q (cos') and cos' 6= t1;1�"k�1+"; then " = 0 (see Remark 5.58), s(C) = k; �(C) = 0;�(C) = 2s(C) � 1, fC(t) = t��1�� T 1;0k�1(t;�)T 1;0k�1(1;�) : In both cases C is a distance-regular codeand fC(t) 2 F>(Q): 113

Degreeof f (�)(t) L(n)Q (�) Intervalfor � = cos'1 1���� [�1;� 1n ]2 2n(1��)1�n� [� 1n ; 0]3 n(2+(n+1)�)(1��)1�n�2 [0; 1pn+3+1 ]4 2n(1+(n+2)�)(1��)1+2��(n+2)�2 [ 1pn+3+1 ; 1pn+2 ]5 n((n+2)(n+3)�2+4(n+2)��n+1)(1��)2�(3�(n+2)�2) [ 1pn+2 ; t1;03 ]Table 6.1: The First Five Bounds on the Size of a Spherical Code with MinimalAngular Distance ' (� = cos')The function L(n)Q (�) is continuous and is expressed by the explicit formula inevery interval considered. The �rst �ve formulas are given in Table 6.1. They areobtained using the optimal (for the restricted �-packing problem) polynomials f (�)(t)of degree one, two, three, four, and �ve, respectively.Example 6.3. Find parameters of a hypothetical spherical code C � Sn�1 withn = 24 and d(C) = 1 (or with the minimal angular distance '(C) = 60�) for which(6.15) is attained. Using Remark 5.58 and the calculations given above we haveconsecutively � = �Q(C) = 12 ; k = kQ(�) = 6; " = "Q(�) = 1; � = t1;15 ; jCj =L(24)Q (t1;15 ) = 2�287 � = 196560; s = s(C) = 6; � = �(C) = 1; � = �(C) = 11 and Cmust be a tight 11-design. Moreover,fC(t) = (t+ 1)Q1;15 (t) = (t+ 1)Q(26)5 (t) = 3245(t� 12)(t� 14)t(t+ 14)(t+ 12)(t+ 1)and hence �(dj); j = 1; :::; 6; are equal to 12 ; 14 ; 0;� 14 ;� 12 ;�1; respectively. The coef-�cients of polynomials f (j)C (t); j = 1; :::; 6; over the system fQ(24)i (t)g (see (3.64) and

114

(3.65)) and the values f (j)C (1) are given in the following table:j fj;0 fj;1 fj;2 fj;3 fj;4 fj;5 f (j)C (1) fj;0jCj � f (j)C (1)1 11468 232819 391252 1702351 2300273 920189 20 46002 28117 19521365 9263 � 5888585 � 9200273 � 147263 �64 471043 3778 191 � 25342 4639 460091 92021 90 931504 28117 � 1184819 9263 2944351 � 9200273 � 7360189 �64 471045 11468 � 76273 391252 � 506117 2300273 92063 20 46006 0 � 14095 0 � 461755 0 � 184189 �1 1By Theorem 3.21 C is distance-regular with distance distribution Bdj (C) = p0j;j ;j = 1; :::; 5; given in the last column of this table (see (3.66), (3.67), and Corollary5.49) and with other intersection numbers pki;j de�ned by (3.68). For example,p32;3 = jCj 5Xl=0 (l + 22)f2;lf3;l(2l + 22)�l+22l �Ql(0)� f (3)C (1) = 22374:The sought code really exists. It consists of vectors of minimal norm in the Leechlattice and is unique up to isometry (see [28, Chapter 14]).The parameters of the known (to the author) codes for which the bound (6.15)is attained are presented in Table 6.2. All these codes are described by Delsarte,Goethals, and Seidel in [35]. An exception is the in�nite sequence of codes C � Sn�1of length n = q(q2�q+1) with cos' = q�2 where d(C) = 2 sin '2 and q is a power of aprime. These codes are obtained with the help of arranging on Sn�1 the point graphof the partial geometry (q2 + 1; q + 1; 1) and were considered by Cameron, Goethals,and Seidel in [27] as spherical 3-designs. The author noticed that these codes aremaximal and is not aware of any other in�nite family of maximal codes on Sn�1with a minimal angular distance less than �=2: Table 6.2 also contains parametersof all known tight designs because, for them, both the bounds (6.14) and (6.16) areattained. Detailed information on the existence of tight spherical t-designs can befound in [10].The two most interesting cases are where equality in the bound (6.15) allow theauthor to �nd two kissing numbersM8 = 240 and M24 = 196560:115

n �(C) s(C) �(C) Tightor not cos'(x; y)x; y 2 C; x 6= y jCj =L(n)Q (d(C)) Commentsn 1 1 1 T �1 2n 2 1 0 T � 1n n+ 1 Regularsimplexn 3 2 1 T �1; 0 2n Generalizedoctahedronq q3+1q+1 3 2 0 � 1q ; 1q2 (q + 1)(q3 + 1) q is a power ofa prime; q � 35 3 2 0 � 35 ; 15 1621 3 2 0 � 27 ; 17 16222 3 2 0 � 411 ; 111 1006 4 2 0 T � 12 ; 14 2722 4 2 0 T � 14 ; 16 2753 5 3 1 T �1;� 1p5 12 Regularicosahedron7 5 3 1 T �1;� 13 5623 5 3 1 T �1;� 15 55222 5 3 0 � 12 ;� 18 ;+ 14 8918 7 4 1 T �1; 0;� 12 24023 7 4 1 T �1; 0;� 13 460024 11 6 1 T �1; 0;� 14 ;� 12 1965602 r � 1 � r2� 1+(�1)r2 T cos 2�jrj = 1; :::; �r2� r Regularr-polygonTable 6.2: The Parameters of the Known Spherical Codes C � Sn�1 for Which theUpper Bound (6.15) Is Attained116

This result was obtained independently by Odlyzko and Sloane [87] who used a com-puter to �nd a solution of the �-packing problem for the system fQ(n)i (t)g when� = 12 = cos �3 and 3 � n � 24: Their study shows that, despite the presence of manycases of attainability, in some cases there exist polynomials of higher degree whichimprove (6.15). Boyvalenkov, Danev and Bumova [20] found necessary and su�cientconditions for the existence of such polynomials (see Theorem 5.47). This situation isnot surprising because the asymptotic bounds of Theorem 6.7 and Corollary 6.8 belowshow that (6.15) can be improved when ' is su�ciently small. It should also be notedthat results of Rankin [91] and Astola [5], obtained with the help of other methods,improve, respectively, the �rst and the second, and the third bounds in Table 6.1 forsome '.Note that (6.15), (6.7), and (6.9) give the following boundsMn � L(n)Q (12); n = 2; 3; ::: (6.17)�n � min�6�'��2 �sin n' L(n)Q (cos 2')� ; n = 2; 3; ::: (6.18)which are better than the known bounds (see [94], [28]) for su�ciently large n:Theorems 6.1 and 6.2 imply that for any spherical (2l��)-design C � Sn�1; wherel 2 f1; 2; :::g and � 2 f0; 1g;�(d(C)) = cos'(C) � t1;�l�� = pn�12 ;n�32 +�l�� (6.19)with equality if and only if C is a tight (2l��)-design. Let �(�(C)) = cos (C) ( (C)can be considered as the angular covering radius of C); and let, for any x 2 Sn�1;fx;C(t) be the polynomial of minimal degree such that fx;C(cos'(x; y)) = 0 for ally 2 C and fx;C(1) = 1:Theorem 6.4. [39]For any spherical (2l � �)-design C � Sn�1; where l 2 f1; 2; :::gand � 2 f0; 1g; �(�(C)) = cos (C) � t0;1��l = pn�32 ;n�12 ��l (6.20)with equality if and only if there exists a point x 2 Sn�1nC such that fx;C(t) =� t+12 �1��Q0;1��l (t): In particular, the bound (6.20) is attained for all tight 2l-designs.Thus, the spherical caps of radial angle (C) � arccospn�32 ;n�12 ��l with centers atthe points of any spherical (2l � �)-design C � Sn�1 cover Sn�1:Corollary 6.5. For any spherical (2l� �)-design C � Sn�1; where l 2 f1; 2; :::g and� 2 f0; 1g; jCj�n�1(arccospn�32 ;n�12 ��l ) � �n�1: (6.21)Recently Yudin [119] obtained the following stronger result.117

Theorem 6.6. [119]For any spherical � -design C � Sn�1;jCj�n�1(arccos pn�12 ;n�12� ) � �n�1: (6.22)The bound (6.22) is attained for n = 2 and any �; similarly to bounds (6.14) and(6.21), and improves both these bounds for su�ciently large �: Combining Theorems6.2 and 6.6, for any spherical � -design C � Sn�1 we haveL(n)Q (cos'(C)) � �n�1(arccos pn�12 ;n�12� ) � �n�1: (6.23)This gives one more bound (compare with (6.19)) on the minimal angular distance ofspherical � -designs.Some known asymptotic results for spherical codes are consequences of the follow-ing special case of the bound (6.16):jCj � 2�k + n� 1k � if 1� (d(C))22 = cos'(C) � t1;1k : (6.24)Since t1;1k = pn�12 ;n�12k ; by Corollary 5.17 we havep2hk � t1;1k r (2k + n� 4)(2k + n� 2)k + n� 2 � 2pk � 1 (6.25)where n � 2 and hk is de�ned by (5.35) (take into account (5.36)).Theorem 6.7 (KL bound [57]). For any �xed ', 0 < ' < �2 ; and n!11n logM(n; ') . 1 + sin'2 sin' log 1 + sin'2 sin' � 1� sin'2 sin' log 1� sin'2 sin' : (6.26)It is natural to compare this bound with the trivial sphere packing bound (2.17)which in the case of Sn�1 takes the formM(n; ') � �n�1�n�1('=2)and gives rise (see (2.20)) to the following asymptotic bound for any �xed ', 0 < ' <�2 ; as n!1 1n logM(n; ') . �12 log (1� cos') : (6.27)The di�erence between the right-hand sides of (6.27) and (6.26) (hereafter we uselog to the base 2) is a positive valued convex function which has a maximum equalto 0.0990 (with a de�ciency) at the point '� being the unique root of the equationcos' ln 1+sin'1�sin' + (1+ cos') sin' = 0: ('� = 63� to within 10 angular sec.) Using theknown inequality (see, for example, [103])�sin '2 �n�1M(n; ') � p2�n�sin 2�n�1M(n+ 1; ) if ' < (6.28)for = '� one can improve (6.27) by 0.0990 for smaller angles.118

Corollary 6.8 ([57]). For any �xed ', 0 < ' � '�; and n!11n logM(n; ') . �12 log (1� cos')� 0:0990 (6.29)and, in particular, Mn =M(n; �3 ) � 2�n(0:401+o(1)) as n!1:Corollary 6.9 ([57]). �n � 2�n(0:599+o(1)) as n!1: (6.30)Note that the �rst asymptotic bound �n � 2�n(c+o(1)) with c > 12 was obtainedby Sidelnikov [103].One can use (6.8) and the asymptotic behavior of L(n)Q (cos 2') as '! 0 to obtainan upper bound for �n which is valid for all n = 1; 2; ::: (similar to the bound (6.18)).Corollary 6.10 ([65]). For any n; n = 1; 2; ::: ,�n � �(n) = �j(n2 )�n�� �n2 + 1��2 4n ; (6.31)where j(�) is the smallest positive zero of the Bessel function J�(z) (see [14]).In proving Corollary 6.10 it was also used that�n�1(')�n�1 � �(n2 )2�(n+12 )�( 12 ) sinn�1 ' as '! 0 (6.32)and that, for any �xed � and � (� ,� > �1),arccosp�; �k � j(�)k as k !1 (6.33)(see [113]). Sincej(v) = � + c�1=3 +O(��1=3) as � !1, where c = 1; 855757, (6.34)(see [122]), we have �(n) � ec(4n)1=3�n � �e4�n as n!1. (6.35)Therefore, (6.31) gives the asymptotic bound�n � 2�n(log2 4e+o(1)) as n!1;119

which is weaker in comparison with (6.30).Since the function L(n)Q (cos') is monotone, we can use it to obtain upper boundson the minimal angular distance '(C) or lower bounds on the maximal inner productcos'(C) of spherical codes C � Sn�1 of a given size. Lets(n;M) = minC�Sn�1;jCj=M cos'(C):From (6.24) and (6.25) we have the following.Corollary 6.11. For any n � 2 and k � 1;s(n; 2�k + n� 1k �) �s 2(n+ k � 2)(n+ 2k � 4)(n+ 2k � 2)hk:Corollary 6.12 ([67]). Let n!1; thens(n;M) &8<: q 4 logMn(logn�log logM) if logMlogn !1; logMn ! 0;2p (1+ )1+2 if logMn ! G( ); 0 < <1;where G(x) = (1 + x)H( 11 + x ) and H(x) = �x logx� (1� x) log (1� x) :Using (6.24), (6.25), (6.32)-(6.34) one can also �nd the asymptotic behavior of thebounds (6.14), (6.22), (6.19), (6.20), and (6.23) for spherical � -designs C � Sn�1 indi�erent processes: when � is �xed and n!1; when � grows with n, and when n is�xed and � !1 (see [67], [39]). In particular, in the latter process the both bounds(6.14) and (6.22) for spherical � -designs C � Sn�1 have the following form:jCj & l(n)�n�1 as � !1with l(n) = 12n�2�(n) and l(n) = 2�(n+12 )�( 12 )�(n2 ) �j(n�12 )�n�1 ;respectively. It is surprising that the ratio of the function l(n) for DGS bound (6.14)to that for Yudin's bound (6.22) is exactly equal to �(n � 1) (The de�nition andasymptotic behavior of �(n) are given in (6.31) and (6.35).)Now we calculate bounds for the projective spaces TmPn�1 (see Examples 2.4,2.11 and (3.77)), which can be used to estimate the size of codes with given maximummodulus of the inner product (2.22) over distinct code points. Let T1 = R, T2 = C,and T4 =H, the quaternionic algebra, and120

TmSn�1 = (u = (u1; :::; un); ui 2 Tm; nXi=1 juij2 = 1) ; (6.36)in particular, RSn�1 = Sn�1. For any code C � TmSn�1 we de�ne the angle &(C),0 � &(C) � �2 , as follows: cos &(C) = maxu;v2C, u6=v j(u;v)j. (6.37)If &(C _) > 0; then any pair of di�erent points u and v of C � TmSn�1 lies on di�erentlines U and V of TmPn�1 and (see (2.24))d(U; V ) =p1� j(u;v)j �p1� cos &(C) = p2 sin &(C)2 .Hence, for any code C � TmSn�1,jCj � A(TmPn�1;p2 sin &(C)2 ) if &(C _) > 0.The compact metric spacesTmPn�1 with the standard substitution �(d) = 2 �1� d2�2�1 (or �(d) = cos 2' if d = p2 sin '2 ) are polynomial and Qi(t) = P�;�i (t), where� = m2 (n� 1)� 1; � = m2 � 1: (6.38)Let L�;�(�) = LQ(�) if Q = nP�;�i (t); i = 0; 1; :::o : From (5.90), (5.96), (5.97), and(2.79) it follows thatL�;�(�) = �k + �k � 1��k+�+�+"k�1+" ��k+��1+"k�1+" � 1� P�+1;�+"k�1 (�)P�;�+"k (�) ! , (6.39)where k = k(�) and " = "(�) (or, equivalently, p�+1;�+1�"k�1+" � � < p�+1;�+"k ); inparticular, for any l = 1; 2; ::: and � 2 f0; 1g, L�;�(p�+1;�+�l�� ) equals (2.79). ByLemma 3.22 the system Q for TmPn�1 satis�es the strengthened Krein conditionand we can use Theorem 5.48.Theorem 6.13 ([66], [68]). For any code C � TmSn�1 with &(C _) > 0;jCj � L�;�(cos 2&(C)); (6.40)and, in particular, for any l = 1; 2; ::: and � 2 f0; 1g,jCj � �l+�+1��l�� ��l+�+�+1l ��l+�l � if cos 2&(C) � p�+1;�+�l�� , (6.41)where �, �, and L�;�(�) are de�ned in (6.38) and (6.39).121

In the �rst interval, where �1 � cos 2&(C) � �1 + 2mmn+2 or 0 � cos &(C) �q mmn+2 , (6.40) coincides with Welch bound jCj � n(1�cos2 &(C))1�n cos2 &(C) . For larger cos &(C)(6.40) improves the Sidelnikov [102] and Welch [123] bounds for real and complexcodes C with both the given cos &(C) and the given periodic and aperiodic crosscorre-lation (see Chapter (Helleseth, Kumar), where (6.40) for the �rst four intervals is alsogiven). By Corollary 5.54 the right-hand side of the �rst inequality in (6.41) is thelower bound (5.145) on the size of (2l � �)-designs in TmPn�1. It was also obtainedin Section 2.2 and included in Table 2.1. Numerous examples of codes with equalityin (6.40) and tight designs in TmPn�1 can be found in [34], [35], [66], [53], [54], [68].They all are distance-regular (see Corollary 5.51 and Remark 5.58).6.2. The Hamming spaceThe Hamming space X = Hnv (n; v = 2; 3; :::) (see Examples 2.1, 2.8, 4.1) consistsof vectors x = (x1; :::; xn) where xi 2 f0; 1; :::; v � 1g with the distance d(x;y) beingequal to the number of coordinates in which x and y di�er. This space is distance-transitive and hence distance invariant with the measure wi = v�n�ni�(v � 1)i; i =0; 1; :::; n. Hnv is a self-dual (systems Q and P coincide) polynomial graph [30] withrespect to the linear standard substitution �(d) = 1 � 2dn : The relationship (3.6) infact proves this fact and shows that the system fQk(t); k = 0; 1; :::; ng may be de�nedby means of Qk(�(d)) = (rk)�1Kn;vk (d) (6.42)where Knk (z) = Kn;vk (z) = kXj=0(�1)j(v � 1)k�j�zj��n� zk � j� (6.43)is the Krawtchouk polynomial of degree k and rk = �nk�(v � 1)k: For the polynomialsKn�a�bk (z � a) where a; b 2 f0; 1g the following orthogonality conditions hold:nXd=0Kn�a�bi (d� a)Kn�a�bj (d� a)(v � 1)dda(n� d)b�nd� = 0if i; j 2 f0; 1; : : : ; n� a� bg and i 6= j (see [68] or [71]). Considering (4.33)-(4.38) forthis case one can �ndc0;1 = v2 ; c1;0 = v2 (v � 1) ; c1;1 = nv24(n� 1)(v � 1) ;r0;1k = �n� 1k �(v � 1)k; r1;0k = � kPi=0 �ni�(v � 1)i�2�n�1k �(v � 1)k ; r1;1k = � kPi=0 �n�1i �(v � 1)i�2�n�2k �(v � 1)k ;122

Q0;1k (�(d)) = Kn�1k (d)�n�1k �(v � 1)k ; Q1;0k (�(d)) = Kn�1k (d� 1)kPi=0 �ni�(v � 1)i ;Q1;1k (�(d)) = Kn�2k (d� 1)kPi=0 �n�1i �(v � 1)i ;and, in particular, prove thatda;bk (Q) = da;bk (P ) = dk(n� a� b) + a; (6.44)where dk(n) = dk(n; v) is the smallest root6 of (6.43). By Lemma 3.25 (see alsoExample 3.23) the system Q for the Hamming space satis�es the strengthened Kreincondition.Let Ln;v(d) = LQ(�(d)) and Mn;v2k�1(d) =M2k�1(�(d))(see (5.83)). Then using (5.84) and (5.96) we haveMn;v2k�1(d) = k�1Xi=0 �ni�(v � 1)i ��nk�(v � 1)kKn�1k�1 (d� 1)Knk (d) (6.45)and Ln;v(d) = � Mn;v2k�1(d) if dk(n� 1) < d� 1 � dk�1(n� 2)vMn�1;v2k�1 (d) if dk(n� 2) < d� 1 � dk(n� 1) (6.46)and, in particular (see (5.97)), for any � 2 f0; 1g and any integer l; 1 � l + � � n;Ln;v(dl(n� 1� �) + 1) = LQ(t1;�l ) = v� lXi=0 �n� �i �(v � 1)i: (6.47)It is useful to introduce the following notation for the right-hand side of (6.47):Hn;v(2l+ 1 + �) = v� lXi=0 �n� �i �(v � 1)i: (6.48)For � = 0 it equals the size of a metric ball of radius l around any point of Hnv :Note that if f(t) and ef(t) are annihilating and dual-annihilating polynomials for acode C � Hnv ; then f(�(z)) and ef(�(z)) are polynomials in z of the same degree(�(d) is a linear function) such that all nonzero distances or dual distances are their6Since t1;10 = �1 and tn+1 = 1 (see (5.89) and (5.2)), and �(d) = 1 � 2 dn ; we should putd0(n) = n+ 1 and dn+1(n) = 0: 123

roots. These polynomials in z are referred to as annihilating for distances or for dualdistances of C; respectively. Such nonzero polynomials of the minimal degree arecalled minimal.The universal bounds for designs and codes in Hamming space are complete be-cause Conjectures 3.26 and 4.20 are true and the strengthened Krein condition issatis�ed. Their symmetry is explained by the coinciding systems Q and P (comparewith Johnson space). The following statements are consequences of Theorems 4.27,5.59, 5.55, 5.63, 5.60 and 5.62.Theorem 6.14 ([107]). For any code C � Hnv ;vd0(C)�1 � jCj � vn�d(C)+1: (6.49)Each of the bounds is attained if and only if d(C) + d0(C) = n + 2: In this casenQi=d(C)(i � d) and nQi=d0(C)(i � d) are annihilating polynomials for distances and dualdistances of C respectively.Codes for which these bounds are attained are called MDS-codes. The Reed-Solomon codes belong to the class. Detailed information about the existence of MDS-codes is contained in [78].Theorem 6.15 ([92], [50]). For any code C � Hnv ;Hn;v(d0(C)) � jCj � vnHn;v(d(C)) : (6.50)Equality in the left-hand side holds if and only if d0(C) = 2s(C) + 1� �(C); in thiscase the polynomial (n� z)�Kn�1��s�� (z � 1) (6.51)with s = s(C); � = �(C) is minimal for distances of C; and C is a distance-regularcode. Equality in the right-hand side holds if and only if d(C) = 2s0(C) + 1� �0(C);in this case the polynomial (6.51) with s = s0(C); � = �0(C) is minimal for dualdistances of C:Codes of size equal to the left-hand side or the right-hand side of (6.50) are calledtight designs or perfect codes respectively (in particular, this means that we considerperfect codes with even minimal distance as well). The problem of existence of perfectcodes is considered in [76], [78] and in Chapter (Honkala, Tietavainen). To simplifysome future formulations we assume that d(C) � 2 and d0(C) � 2 .Theorem 6.16 ([64], [71]). Let C be a code in Hnv ; � = �(d(C)); k = k(�); " ="(�); �0 = �(d0(C)); k0 = k(�0); "0 = "(�0) and hence dk(n � 1 � ") < d(C) � 1 �dk�1+"(n� 2 + "); dk0 (n� 1� "0) < d0(C)� 1 � dk0�1+"0(n� 2 + "0): ThenvnLn;v(d0(C)) � jCj � Ln;v(d(C)) (6.52)124

where Ln;v(d) is de�ned by (6.46) and, in particular,jCj � Hn;v(2l+ 1 + �) if d(C) � dl(n� 1� �) + 1 (6.53)jCj � vnHn;v(2l0 + 1 + �0) if d0(C) � dl0(n� 1� �0) + 1: (6.54)The upper bound in (6.52) is attained if and only ifd0(C) � 2s(C)� �(C):Herewith, if jCj = Ln;v(d(C)) and d(C) = dk�1+"(n � 2 + ") + 1, then s(C) = k;�(C) = 1�"; �(C) = 2s(C)��(C) (and hence C is a tight design), and the polynomial(n� z)�Kn�1��s�� (z � 1) (6.55)with s = s(C); � = �(C) is minimal for distances of C; if jCj = Ln;v(d(C)) andd(C) 6= dk�1+"(n� 2+ ")+1, then s(C) = k+ "; �(C) = "; �(C) = 2s(C)��(C)� 1and the polynomial(z � d)(z � n)� s�1Xi=0 Kn�1��i (d� 1)Kn�1��i (z � 1)�n�1��i �(v � 1)i (6.56)with s = s(C); � = �(C); d = d(C) is minimal for distances of C. In the both casesC is a distance-regular code. The lower bound in (6.52) is attained if and only ifd(C) � 2s0(C)� �0(C):If jCjLn;v(d0(C)) = vn and d0(C) = dk0�1+"0(n�2+ "0)+1, then s0(C) = k0; �0(C) =1 � "0; d(C) = 2s0(C) + 1 � �0(C) (and hence C is a perfect code), and (6.55) withs = s0(C); � = �0(C) is minimal for dual distances of C; if jCjLn;v(d0(C)) = vn andd0(C) 6= dk0�1+"0(n�2+"0)+1, then s0(C) = k0+"0; �0(C) = "0; d(C) = 2s0(C)��0(C)and the polynomial (6.56) with s = s0(C); � = �0(C); d = d0(C) is minimal for dualdistances of C.The values Ln;2(d); that give rise (see (6.52)) to the �rst �ve bounds for binarycodes and designs, are given in Table 6.3. The �rst three bounds for codes are thesame as in the case of the Euclidean sphere. This is explained by the coincidenceof polynomials Qi(t); i = 0; 1; 2: The �rst bounds for codes and designs belong toPlotkin [89] and Friedman [41], respectively. The third bound for binary codes, insome range, was improved by Tiet�av�ainen [116] with the help of a special method (seealso [60]). At the same time, by Theorem 6.16 the general third boundjCj � vd ((n(v � 1) + 1) (n(v � 1)� vd+ 2)� v)vd (2n(v � 1)� v + 2� vd)� n(n� 1)(v � 1)2 ,125

Degreeof f (�)(t) LQ(�) = Ln;2(d)for � = 1� 2 dn Intervalfor � = 1� 2 dn1 1���� [�1;� 1n ]2 2n(1��)1�n� [� 1n ; 0]3 n(2+(n+1)�)(1��)1�n�2 [0; pn�1�1n ]4 2n(n�2+�n2)(1��)n�1�(n��1)2 [pn�1�1n ; pn�2n ]5 (4n3�+n2(n2+n+2)�2�(n�1)(n�2)2)(1��)2�(3n�2�n2�2) [pn�2n ; p3n�5�1n ]Table 6.3: Values LQ(�) = Ln;2(d); � = 1� 2 dn , for the First Five Intervalswhere d2(n � 1) + 1 � d = d(C) < d1(n � 2) + 1, can be attained only for distance-regular 3-designs with two distances d and (v�1)(n�1)v �1 + 1(v�1)(n�1)�v(d�1)� + 1;there exist in�nite families of such codes (see [24] and Table 6.4).The parameters of the known (to the author) codes for which the upper bound in(6.52) is attained are presented in Table 6.4. This class of codes consists of distance-regular codes and includes all tight designs. The lower bound in (6.52) is attained forcodes with dual parameters (in particular, for all perfect codes), if they exist.For the Hamming space Conjectures 3.26 and 4.20 are completely proved.Theorem 6.17 ([29], [71]). For any code C � Hnv ;vnHn;v(2s0(C)) + 1� �0C)) � jCj � Hn;v(2s(C) + 1� �(C)): (6.57)The conditions for attainability of the lower and upper bound in (6.57) coincide withthose for the upper and lower bounds in (6.50) respectively.All statements of Theorem 4.5 are consequences of universal bounds (6.49), (6.50),(6.52), (6.57). Moreover, together with the inequality d(C)+d0(C) � n+2 they implytwo other inequalitiesHn;v(d0(C))Hn;v(d(C)) � vn; Ln;v(d0(C))Ln;v(d(C)) � vnwhich show that d(C) and d0(C) cannot be too large simultaneously. The last inequal-ity is stronger and gives rise to an improvement of some bounds on minimal distanceof self-dual codes [71].The following result of Tiet�av�ainen [117], [118] can be considered as a special caseof Theorem 4.14.Theorem 6.18 (Tiet�av�ainen bound). If for a code C � Hnv ; d0(C) = 2l + 1 � �for some l 2 f1; 2; :::g and � 2 f0; 1g; then�(C) � dn;vl (n� 1 + �): (6.58)126

n v s(C) �(C) d0(C) d(C) = d jCj = Ln;v(d) Commentsn v 1 1 2 n vn v 1 0 2 n > d >(v�1)n+1v vdvd�n(v�1) coexistence [99]with resolvableblock-designs2� (L;L=v; n� d)n v 1 0 3 (v�1)n+1v (v � 1)n+ 1 coexistence [100]with a�neresolvableblock-designs2� (L;L=v; n� d)plq q = pm 2 1 3 n� pl nq l;m = 1; 2;Semakov, Zinovjev,Zaitzev [100]qh+h� q q 2 1 3 n� h q3 2jq; hjq; 2 < h < qDenniston [37]q2 + 1 q 2 0 4 n� q � 1 q4 ovoid in PG(3; q)Bose [19], Qvist [90]56 3 2 0 4 36 36 projective capHill [51]78 4 2 0 4 56 46 projective capHill [52]4l 2 2 1 4 n=2 2n Hadamard codesq + 2 q 2 1 4 n� 2 q3 2jq hyperovalin PG(2; q)Bose [19]11 3 2 0 5 6 35 projectionof Golay code12 3 3 1 6 6 36 Golay code [49]22 2 3 0 6 8 210 projectionof Golay code23 2 3 0 7 8 211 projectionof Golay code24 2 4 1 8 8 212 Golay code [49]n 2 bn=2c 1+(�1)n2 n 2 2n�1 even weight codeTable 6.4: The Parameters of the Known Codes C � Hnv for Which the Upper Bound(6.52) Is Attained127

In [39] it was proved that equality in (6.58) holds if and only if there exists apoint x 2 Hnv nC such that the polynomial (n � z)1��Kn�1��;vl (z) is minimal for alldistances d(x; y); y 2 C; and that in the binary case the bound (6.58) is attained forall tight 2l-designs.Some asymptotic results are consequences of the following special case of the bound(6.53) jCj � kXi=0 �ni�(v � 1)i if d(C) � dk(n� 1) + 1 (6.59)and estimates for dk(n) (see Corollaries 5.20 and 5.21). In particular, we have thefollowing result of McEliece, Rodemich, Rumsey Jr., and Welch [79].Theorem 6.19 (The �rst form of the MRRW bound). If v is �xed and limn!1dn = � where 0 � � � v�1v ; then1n logv A(Hnv ; d) . Hv( v(�)) (6.60)where Hv(x) = �x logv x� (1� x) logv (1� x) + logv (v � 1) ; (6.61) v(x) = 1v �v � 1� (v � 2)x� 2p(v � 1)x(1� x)� : (6.62)Notice that v(x); 0 � x � (v � 1)=v, is a decreasing continuous function which,as one can check, coincides with its inverse function, that is v( v(x)) = x when 0 � x � v � 1v :Using (6.54) one can obtain the following result.Theorem 6.20 ([71]). If v is �xed and limn!1 dn = � where 0 � � � v�1v ; then1n logv B(Hnv ; d) & 1�Hv( v(�)): (6.63)Similar to the Euclidean case the asymptotic bound (6.60) (and also (6.63)) is notgood for small �: In this case it becomes even worse than the trivial sphere packingbound (the upper bound in (6.50)) which in the stated asymptotic process takes theform 1n logv A(Hnv ; d) . 1�Hv(�): (6.64)In the binary case the following inequality (Bassalygo-Elias Lemma [11])�nw�A(Hn2 ; 2d) � 2nA(Jnw ; d) (6.65)128

is valid. This plays the same role as (6.28) and allows us to improve (6.60) for small� using the universal bound for codes in Johnson space. In the non-binary case thesituation is more di�cult because a natural generalization of (6.65) connects Hnv witha metric space which is not polynomial (see [106], [1], [114]).Similar di�culties arise for studying codes in eHnv = nu = 1pn (u1; :::; un); ui 2 eHvo,where eHv is the set of all v-th roots of unity. Since eHnv � CSn�1 (see (6.36)), onecan consider the problem of �nding the maximum size of a code C � eHnv with a givenvalue & = &(C) (see (6.37)), which is of important interest for codes with given cross-correlation properties. However, the problem to describe all polynomials f(t) suchthat f(j(u;v)j2) is an FDNDF on eHnv , v � 3, is solved now (1997) only for polyno-mials of small degrees (the corresponding �rst three bounds can be found in [66] andChapter (Helleseth, Kumar). At the same time, for v = 2, 2n�1 pairs U = fu;� ug ofcomplementary points of eHn2 (lines of \projective Hamming space") with the distanced(U; V ) = minu2U;v2V d(u;v) form a polynomial graph (called a folded cube [23])with D = bn2 c and the (standard for even n) substitution �(d) = 2(1� 2dn )2 � 1. Thecorresponding inequality jCj � LQ(cos 2&(C)) if &(C _) > 0 (cf. (6.40)) was proved andcalculated in [69]. This improves the Sidelnikov bound [102] for binary codes whencos &(C _) > p3n�8n . The �rst four bounds are also given in [69] and Chapter (Helleseth,Kumar).6.3. The Johnson spaceWe consider Johnson space Jnw (n = 2; 3; ::: ;w = 1; :::; bn=2c) as the set X of allw-subsets of the n-set f1; :::; ng ; where the distance between two elements x; y 2 Jnwis de�ned to be w � jx \ yj (see Examples 2.2, 2.9). Jnw can also be considered asthe subset of the binary Hamming space Hn2 consisting of all vectors which haveexactly w non-zero coordinates, with distance being equal to half of the Hammingdistance. For this reason codes in Jnw are often called constant weight codes. In thiscase �(X) = f0; 1; :::; wg and hence s(X) = w; D(X) = w: As shown by Delsarte[30], Johnson space is a polynomial graph withvi = � wi �� n� wi � ; i = 0; 1; : : : ; w; (6.66)ri = � ni ��� ni� 1 � ; i = 1; : : : ; w (r0 = 1); (6.67)with the standard substitutions (see Example 4.2)�Q(d) = 1� 2 dw ; �P (d) = 1� 2 d(n+ 1� d)w(n+ 1� w) : (6.68)The corresponding systems fQi(t); i = 0; 1; :::; wg and fPi(t); i = 0; 1; :::; wg may bede�ned by 129

Qi(�Q(d)) = Ji(d); i = 0; 1; :::; w; (6.69)where Ji(z) = Jn;wi (z) = iXj=0(�1)j �ij��n+1�ij ��wj ��n�wj � �zj� (6.70)are Hahn polynomials, and byPi(�P (d)) = (vi)�1Ei(d); i = 0; 1; :::; w; (6.71)whereEi(z) = En;wi (z) = iXj=0(�1)i�j � w � ji� j �� w � zj �� n� w + j � zj � : (6.72)(We shall verify that (6.72) is really a polynomial of degree i in �P (z):) By Lemma 3.25(see also Example 3.23) the system Q for the Johnson space satis�es the strengthenedKrein condition. Unfortunately this question with respect to the system P is stillopen (1997).A set C is a simple � -design in Jnw with respect to a linear substitution �Q(d) ifand only if (see [30]) it forms a block �-design or tactical con�guration. This object isde�ned as a set C of w-subsets called blocks a of a n-set such that, for some �, each� -subset of the n-set belongs to exactly � w-subsets of C and denoted by S�(�; w; n).In other words, S�(�; w; n) is a binary code consisting of words of length n with wones such that in any � positions the word of all ones occurs the same number �times. The systems S�(�; w; n) with � = 1 are called Steiner systems and denoted byS(�; w; n). Weighted � -designs in Jnw were also considered when C is a multiset and� is the total multiplicity of the w-subsets of C containing a �xed � -subset. Since a� -design is an i-design for any i; 0 � i � �; we have the following necessary conditionsfor the existence of a � -design:��n� i� � i� � 0 �mod �w � i� � i�� ; i = 0; 1; :::; �: (6.73)By Wilson's theorem [124], for any integers n;w; and � , 0 � � � w � n, there exists a(in general, weighted) system S�(�; w; n) for all su�ciently large integers � satisfyingthe congruencies (6.73). We emphasize once more that by Theorem 2.15 the lowerbounds given below are already valid for the number of di�erent blocks of a � -design.The following statement follows from Theorem 4.27 if one takes into account thatby (4.68), (4.70), (6.66), and (6.68),NQ(d) = wXj=0� n� wj �� d� 1j � = � n� w + d� 1n� w � :130

Theorem 6.21. � nd0(C)�1�� wd0(C)�1� � jCj � � nw+1�d(C)�� ww+1�d(C)� : (6.74)Each of the bounds is attained if and only if d(C) + d0(C) = w + 2: In this case C isa Steiner system, and nQi=d(C)(i� d) and nQi=d0(C)(i� d) are annihilating polynomials fordistances and dual distances of C; respectively.There exists a simple combinatorial proof of (6.74). Complete information aboutthe existence of Steiner systems can be found in [16].Hahn polynomials (6.70) and the corresponding polynomials Qi(t) de�ned by(6.69) satisfy the following conditions of orthogonality and normalization:ri wXd=0 Ji(d)Jj(d)vd = �i;j�nw�;Qk(1) = Jk(0) = 1; k = 0; 1; :::; w:In particular, J0(d) = 1;J1(d) = 1� ndw(n� w) ; J2(d) = 1� 2(n� 1)dw(n � w) + (n� 1)(n� 2)d(d� 1)w(w � 1)(n� w)(n � w � 1) :Using (6.67) - (6.70) and Theorem 5.3 one can obtain the recurrence (5.49) for poly-nomials Qi(t): This recurrence implies by induction thatQk(�1) = Jk(w) = (�1)k �wk��n�wk � ; k = 0; 1; :::; w: (6.75)Moreover, for k = 0; 1; :::; w; we have the following equalities (see (5.14) and (5.24)):Tk(1; 1) = kXi=0 ri = �nk�; (6.76)Tk(1;�1) = kXi=0 riQi(�1) = Qk(�1)�nk�w � kw ; (6.77)Rk(1; 1;�1) = Qk(�1)�nk� kw : (6.78)Using Lemma 5.24, (6.67), (6.76)-(6.78) we �nd that in this case the polynomialsQ1;0k (t) and Q0;1k (t) for k � w � 1, and Q1;1k (t) for k � w � 2 are de�ned as follows:Q1;0k (�Q(d)) = (w � k)(n� w � k)n� 2k � Jk(d)� Jk+1(d)d ; (6.79)131

Q0;1k (�Q(d)) = wn� 2k � (w � k)Jk(d) + (n� w � k)Jk+1(d)w � d ; (6.80)Q1;1k (�Q(d)) = w(n� w � k)(n� k)(n� 2k) � w(w � k)Jk(d)� (w(w � k)� d(n� 2k))Jk+1(d)d(w � d) (6.81)(the values Q1;0k (1) = Q1;1k (1) = 1 and the values Q0;1k (�1) and Q1;1k (�1) coincidewith the corresponding limits of the right-hand sides when d tends to 0 or w.) At thesame time Example 3.23 and Theorem 3.24 show thatQ0;1k (�Q(d)) = Jn�1;w�1k (d) (6.82)and, in particular, kXi=0 r0;1i = �n� 1k �; k = 0; 1; :::; w � 1: (6.83)Let da;bk (U) = da;bk (U; n; w) for U = Q and U = P . Since in the general case (see (5.2)and (5.89)) t1;0N�1 = t1;1N�2 = tN;N�1 and t1;10 = �1, we have d1;0w (U) = d1;1w�1(U) = 1and d1;10 (U) = w:From (6.82) it follows thatd0;1k (Q;n;w) = dk(Q;n� 1; w � 1):We shall use that d1;01 (Q;n;w) = w(n�w)n�1 ; d1;11 (Q;n;w) = (w�1)(n�w)n�2 ; andd1;02 (Q;n;w) = (n�2)(2w(n�w)�n�1)�p(n�2)(4w(n�w)(w(n�w)�3n+5)+(n�2)(n+1)2)2(n�2)(n�3) :Assuming for the Johnson space LQ(�Q(d)) = Ln;w;2(d) and using (5.83), (5.85),(5.95), (6.76), (6.79), and (6.82) one can �nd that for 1 < d � w;Ln;w;2(d) = ( M (d)2k�1 if d1;0k (Q;n;w) < d � d1;1k�1(Q;n;w);M (d)2k if d1;1k (Q;n;w) < d � d1;0k (Q;n;w); (6.84)whereM (d)2k�1 = �1� (w � k + 1)(n� w � k + 1)n� 2k + 2 Jn;wk�1(d)� Jn;wk (d)dJn;wk (d) �� nk � 1�;M (d)2k = 1� (w � k)(n� w � k)n� 2k Jn;wk (d)� Jn;wk+1(d)dJn�1;w�1k (d) !�nk�:Using (5.97), (6.75), (6.76), and (6.83) one can also �nd the following special valuesof (6.84) for d = d1;�k (Q;n;w) where � 2 f0; 1g and 1 � k + � � w :Ln;w;2(d1;�k (Q;n;w)) = �1� 1Q1(�1)�� kXi=0 r0;�i = �nw�� �n� �k �: (6.85)132

The function (6.72) is a polynomial of degree i in �P (z) since(w � z � l)(n+ 1� z � w + l) = �z(n+ 1� z) + (w � l)(n+ 1� w + l) (6.86)for any l = 0; 1; :::; j: The polynomials Pi(t) de�ned by (6.71) satisfy the followingconditions of orthogonality and normalization:vi wXd=0Pi(�P (d)Pj(�P (d))rd = �i;j�nw�; (6.87)Pi(1) = (vi)�1Ei(0) = 1; i = 0; 1; :::; w:From (6.66), (6.71), (6.72), (6.86), and Theorem 5.3 it follows that for these polyno-mials Pi(t) the recurrence (5.7) holds wherebi = Co(Pi(t))Co(Pi+1(t)) = 2(w � i)(n� w � i)w(n+ 1� w) ; (6.88)ci = vi�1vi bi�1 = 2i2w(n+ 1� w) ; ai = 2i (n� 2i)� w(n� w � 1)w(n+ 1� w) :Now we calculate LP (t1;�k (P )) = �1� 1P1(�1)�� kXi=0 v0;�i : (6.89)By (5.64) and (6.87) v0;1i = (Ti(1;�1; P ))2�c0;1(P )vibiPi(�1)Pi+1(�1) : (6.90)From (6.71) and (6.72) it follows thatPi(�1) = (vi)�1Ei(w) = (�1)i�n�wi � ; i = 0; 1; :::; w; (6.91)and, hence,Ti(1;�1; P ) = iXj=0 vjPj(�1) = iXj=0�wj�(�1)j = (�1)i�w � 1i �: (6.92)Using (4.34), (6.68), and the fact thatd(n+ 1� d)��nd��� nd� 1�� = n(n� 1)��n� 2d� 1���n� 2d� 2�� (6.93)one can �nd that c0;1(P ) = n+ 1� w2 : (6.94)133

From (6.88)-(6.92) and (6.94) it follows that1� 1P1(�1) = n� w + 1; v0;1i = 1i+ 1�w � 1i ��n� wi �and, hence, LP (t1;�k (P )) = kXi=0 �w � �i ��n� w + �i+ � �: (6.95)Note that from (4.36) and (6.93) it also follows that for 0 � i < j � w�1 the equalitywXd=0P 1;0i (�P (d))P 1;0j (�P (d))(1� �P (d))rd = 0implies thatw�1Xd=0 P 1;0i (�P (d+ 1))P 1;0j (�P (d+ 1))��n� 2d ���n� 2d� 1�� = 0 .Therefore, P 1;0i (�P (d+1)) is equal to En�2;w�1i (d) up to a constant factor and henced1;0k (P; n; w) = dk(P; n� 2; w � 1) + 1: (6.96)Consider codes C � Jnw with d(C) > 1 and d0(C) > 1: It is clear that for any d;d = 2; :::; w; there exist unique integers l and � 2 f0; 1g such that d = 2l + 1 � �.Moreover, for any d; d = 2; :::; w; and U (U = Q or U = P ) there exist minimalpossible integers l and � 2 f0; 1g such that d � d1;�l��(U) (since, by (5.90), d1;"k (U) <d � d1;1�"k�1+"(U); where k = kU (�U (d)) and " = "U (�U (d)); indeed, we have l = k,� = 1� " if d = d1;1�"k�1+"(U) and l = k + ", � = " if d 6= d1;1�"k�1+"(U)).The following universal bounds follow from Theorems 5.59, 5.55, 5.63. For oddd0(C) and d(C) they coincide with the Ray-Chaudhuri-Wilson bound [93] for blockdesigns and the sphere packing bound for codes in Jnw, which are also proved by acombinatorial method.Theorem 6.22. For a code C � Jnw; let d(C) = 2l + 1� � and d0(C) = 2k + 1� #for some l; k 2 f1; :::; wg and �; # 2 f0; 1g: Then� nw�#�n� #k � #� � jCj � �nw�Pl��i=0 �w��i ��n�w+�i+� � : (6.97)Equality in the left-hand side holds if and only if s(C) = k and �(C) = #; in this casefC(t) = � 1+t2 ��(C)Q1;�(C)s(C)��(C)(t). Equality in the right-hand side holds if and only ifs0(C) = l and �0(C) = �; in this case efC(t) = � 1+t2 ��0(C) P 1;�0(C)s0(C)��0(C)(t):134

Theorems 5.60, 5.55, 5.63 give rise to the following result.Theorem 6.23. For a code C � Jnw; let d(C) � d1;�l��(Q) and d0(C) � d1;#k�#(P ) forsome l; k 2 f1; :::; wg and �; # 2 f0; 1g: Then�nw�Pk�#i=0 �w�#i ��n�w+#i+# � � jCj � �nw�� �n� �l � ��: (6.98)Equality in the right-hand side holds if and only if d0(C) = 2s(C)� �(C) + 1; in thiscase fC(t) = � 1+t2 ��(C)Q1;�(C)s(C)��(C)(t). Equality in the left-hand side holds if and onlyif d(C) = 2s0(C)� �0(C) + 1; in this case efC(t) = � 1+t2 ��0(C) P 1;�0(C)s0(C)��0(C)(t):Example 6.24. In the case of the known constant weight code C with parametersn = 24; w = 8; jCj = 759; d0(C) = 6; d(C) = 4 we have d1;12 (Q) = 4; d1;11 (P ) = 6 andall six bounds (6.74), (6.97), and (6.98) are attained!The upper bounds in (6.98) are special cases of the following statement (see The-orems 5.55 and 5.62).Theorem 6.25 ([64], [67]). For any code C � Jnw;jCj � Ln;w;2(d(C)) (6.99)where Ln;w;2(d) is de�ned by (6.84). Equality holds if and only ifd0(C) � 2s(C)� �(C) > 1:If d0(C) = 2s(C)+1��(C) (and hence C is a tight design), then fC(t) = � 1+t2 �� Q1;�s��(t);if d0(C) = 2s(C)��(C), then fC(t) = t��1�� � t+12 �� T 1;�s�1(t;�)T 1;�s�1(1;�) where s = s(C); � = �(C)in both cases and � = �Q(d(C)):Note that a code C � Jnw for which jCj equals the lower bound in (6.97) or theupper bound in (6.99) (in particular, in (6.98)) is distance-regular, and the polynomialfC(�Q(d)) in d of degree s(C) is minimal for distances of C: Theorem 6.25 gives riseto the following bounds for a code C � Jnw with large minimal distance d(C) = d :jCj � dndn� w(n� w) if w(n� w)n� 1 < d � w; (6.100)jCj � dn(n� 1)w(d(n � 1)� (w � 1)(n� w)) if (w � 1)(n� w)n� 2 < d � w(n� w)n� 1 ;jCj � dn(n� 1)(w(n� w) � 1� d(n� 2))d(n� 1)(2w(n� w)� n� d(n� 2))� w(n� w)(w � 1)(n� w � 1)135

if d1;02 (Q;n;w) < d � (w�1)(n�w)n�2 :The �rst of the bounds is the well known Johnson bound [55]. It should be notedthat the bounds of Theorems 6.23 and 6.25 become stronger than those of Theorems6.21 and 6.22 when the parameters w; d(C) and d0(C) are su�ciently large.To derive asymptotic results one can use the following special cases, � = 0 and# = 0 in Theorem 6.23: jCj � �nl� if d(C) � d1;0l (Q); (6.101)jCj � �nw�Pki=0 �wi ��n�wi � if d0(C) � d1;0k (P ): (6.102)Consider for any �; 0 � � � 12 ; the decreasing continuous function��(x) = �(1� �)� x(1� x)1 + 2px(1� x) ; (6.103)which maps the interval [0; �] onto [0; �(1 � �)]. The inverse function ��1� (x) can beexpressed in the following explicit form:��1� (x) = 12 1�r1� 4�p�(1� �)� x(1� x) � x�2! : (6.104)This fact is essentially used to obtained asymptotic results for the Johnson space(compare with the function v(x) for the Hamming space that coincides with itsinverse, see (6.62)). First we note that by Corollary 5.26, Lemma 5.29, Theorem 5.12,and (5.54), tk(Q)� 2nw(n�2k+1) < t1;0k (Q) < tk(Q) and, hence,dk(Q) < d1;0k (Q) < dk(Q)� n(n� 2k + 1) ; k = 1; :::; w � 1: (6.105)Corollary 5.22, (6.101), and (6.103)-(6.105) imply the following known result.Theorem 6.26 ([79]). If limn!1 wn = � and limn!1 dn = �; where 0 < � � 12 and0 � � � �(1� �); then 1n logA(Jnw; d) . H2 ���1� (�)� ; (6.106)where H2(x) is the binary Shannon entropy (see (6.61)).On the other hand, (6.102), (6.96), (6.103), (6.104), and Corollary 5.23 give riseto a new result. 136

Theorem 6.27. If limn!1 wn = � and limn!1 dn = �; where 0 < � � 12 and 0 � � � �;then 1n logB(Jnw; d) & H2(�) � �H2(��(�)� )� (1� �)H2(��(�)1� � ): (6.107)We note for comparison that in the same asymptotic processes Theorem 6.22implies the following bounds:1n logA(Jnw ; d) . H2(�)� �H2( �2� )� (1� �)H2( �2(1� �) );1n logB(Jnw; d) & H2(�2 ):From Theorem 6.26 and inequality (6.65) one can derive the following asymptoticbound for codes in binary Hamming space.Theorem 6.28 (The second form of the MRRW bound [79]). If limn!1 dn = �; where 0 � � � 2�(1� �); then1n logA(Hn2 ; d) . 1� max12 (1�p1�2�)��� 12 �H2(�) �H2���1� (�2)�� ; (6.108)where ��1� (x) is de�ned by (6.104).This asymptotic bound is better than (6.60) for v = 2 when � < 0:272: Note thatin the proof of Theorem 6.28 instead of (6.65) the following signi�cant relationshipbetween linear programming bounds for codes in Hamming and Johnson spaces�nw�AQ(Hn2 ; 2d) � 2nAQ(Jnw; d) (6.109)may be used. This was found by Rodemich (see [33]). New lower bounds on B(Hn2 ; d)and, in particular, an analog of the asymptotic inequality (6.108) were obtained in[72] with the help of (6.109) and Corollary 4.26 applied to the both spaces Hn2 andJnw.References[1] M. J. Aaltonen, \A new bound on nonbinary block codes", Discrete Mathemat-ics, 83, pp. 139-160, 1990.[2] N. Alon, O. Goldreich, J. Hastad, R. Peralta, \Simple construction of almostk-wise independent random variables," Random Structures and Algorithms, 3,no. 3 (1992), 289-304. 137

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