448 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO.2, MARCH 1995

New Constructions of Optimal Cyclically Permutable Constant Weight Codes

Oscar Moreno, Member, IEEE, Zhen Zhang, Senior Member, IEEE, P. Vijay Kumar, Member, IEEE, and Victor A. Zinoviev

Abstract-Three new constructions for families of cyclic con­stant weight codes are presented. All are asymptotically optimum in the sense that in each case, as the length of the sequences within the family approaches infinity, tbe ratio of family size to the maximum possible under the Johnson upper bound, approacbes unity.

Index Terms-CDMA, cyclic constant weight codes, cyclically permutahle codes, optical orthogonal codes, protocol sequences, correlation, crosscorrelation.

I. INTRODUCTION

AN (n, w,)') binary cyclically permutable constant weight

code (we call it here a CPCW code) C, where 1 ::::: ), ::::: w ::::: n, is a family of {n, 1 }-sequences of length n and

Hamming weight w satisfying the following two conditions:

n-l L x(k)x(k El7" T) ::::: ), (1) k=O

for all sequences xC) E C and all integers T ¢ n (mod n) and

n-l Lx(k)y(kfIlnT):::::), (2) k=O

for all pairs of distinct sequences x(,), yO E C and all

integers T, where El7n denotes addition modulo n. Codes with these propcrties have been called optical orthog­

onal codes in papers [1]-[3J in connection with applications

for optical channels and cyclically permutable constant wcight

codes (see [4] and references there) in connection to con­

structing protocol sequences for the multiuser collision channel

without fecdback. We use the latter terminology because it fits

better the general results given here, which are not restricted to applications for optical communication channels.

Manuscript received May 21,1993; revised January 4,1994. This research was supported in part by the National Science Foundation under Grants RI!-9014056, NCR-890505, and NCR-9016077, by the Office of Naval Research under Grant NOOOI4-90-J-1301, and by the Computational Mathematics Group of the EPSCoR of Puerto Rico Grant. This paper was presented at the IEEE International Symposium on Information Theory, San Antonio, TX, Jan. 17-22, 1993.

O. Moreno is with the Department of Mathematics, University of Puerto Rico, Rio Piedras, PR 00931 USA

V. A. Zinoviev is with the Department of Mathematics, University of Puerto Rico, Rio Piedras, PR 00931 USA, and with the Institute for Problems of Information Transmission, Russian Academy of Sciences, GSP-4, Moscow, 101447, Russia.

Z. Zhang and P. V. Kumar are with Communication Sciences Institute, University of Southern California, Los Angeles, CA 90089-2565 USA.

IEEE Log Number 9406688.

For a given set of values of n, w, A, let <l>(n,w, A), denote the largest possible cardinality of an (n, w, ),) CPCW

code. Upper bounds for this function and several optimal

constructions can be found in [1]-[41. An easy upper bound derived from the Johnson bound A(n,2(w - ),),w) (see [5]) statcs that

<I>(n,w,),)::::: l A(n,2(W

n-),),w)J

< (n-l)(n-2)",(n -),), (3) - w(w - 1) . . · (w - A)

Constant weight codes, which are optimal relative to this

Johnson upper bound, are also of interest from the combina­

torial point of view (see [6] and references there). In this paper, three constructions (A, B, and C) for families

of CPCW codes are presented. In every case, the families ,are

asymptotically optimum in the sense that, as the length of the

sequence family goes to infinity, the ratio of the size of the code to that of the maximum permissible as determined by the

bound in (3) above, approaches unity.

All three constructions make use of the following two ideas,

Let n be an integer that can be expressed as the product n =

'f!1T!2 of two relatively prime integers nl and n2. Then, from an application of the Chinese remainder theorem, it follows

that the construction of sets of {O, 1} sequences with periodic

correlation bounded above by ), is completely equivalent to the task of constructing a collection of arrays whose doubly­

periodic correlation is bounded above by A. Secondly, the

codewords within each family are required to have constant

weight. The sequences in each of the three families A, B, and

C when represented in matrix form appear as the graph of a function mapping Zn2 -> Z"l' This guarantees that they all

have constant weight (approximately) 1/,2. The functions in A and B are polynomials, whereas construction C uses rational

functions, Precise parameters of the three families constructed are

tabulatcd below. Reference l4J appeared after we had written

the initial version of this paper. The two papers share some material in common such as the idea behind the construction

as well as some featurcs of construction A. On the other hand,

Section III of this paper was written after paper [4], and was

inspired by it.

II. CONSTRUCTIONS

The three constructions that follow depend upon the fol­

lowing observation:

0018-9448/95$04.00 (Q) 1995 IEEE

MORENO et al.: NEW CONSTRUCTIONS OF OPTIMAL CYCLICALLY PERMUTABLE CONSTANT WEIGHT CODES 449

Let A = [A(i,j)] and B = [B(i,j)] be rxs matrices having {O, I} entries where rand s are relatively prime. Let at-) and b(·) be the sequences of length rs associated with the matrices A and B respectively, via the Chinese remainder theorem, i.e.,

a(l) = A(I mod (1') , I mod (8»)

and similarly,

b(l) = B(l mod ('1') , I mod (8») for all l, 0 ::; l ::; rs - 1. It follows from the linearity of the residue map, that for any value of T, 0 ::; T ::; rs -1

rs-l r-l s-1 L all $1"S T)b(l) = L L A(-i. $1' T,j EBs T)B(i,j)

i=Oj=O where EBm denotes addition modulo m. As a result, the col­lection of one-dimensional periodic auto- and cross-correlation values of a family of sequences of length rs is precisely the same as ·the set of two-dimensional doubly-periodic auto- and cross-correlation values of the collection of 'I' x 8 matrices associated with these sequences via the residue map as in the discussion above.

Accordingly, the constructions in this paper present r x s matrices, (r, s are relatively prime) with periodic auto- and cross-correlation bounded above by>..

A second property common to all the constructions is that each r x s matrix is regarded as the graph of the values of a function f from Z, -> Z1' where Zm denotes the set of integers modulo m. More specifically, the (i,j)th entry of the matrix

A(i .)={I, iff(j)=r-I-i

,J 0, otherwise.

As a result, each matrix will have in general s l's, which ensures that the corresponding '('Ii-length sequence has constant weight s.

A. Family A Let p be a prime, m any divisor of p -1, and t an integer,

1 ::; t ::; m - 1. Consider the set of all polynomials over the finite field Fp of p elements of the form

f(:I:) = f8:1;' + fs_l:l:s-1 + . . . + fo (4)

where 1 ::; s ::; t and the coefficients Ii E F p for all i. We would like to discard from this set all those polynomials hex) for which there exist u E F p and v E F; such that

h(:l;) = h(v:I;) + 11"

Clearly, this can happen if and only if (iff) hex) and h(vx)+11, are identical as polynomials. In turn, this is possible with v =1= 1 iff

n=Oand(d,p-l» I

corresponding to nonzero coefficients is relatively prime to p - 1.

The number of elements in F can be counted with the help of the Moebius function

IFI = L (p[(Hl)/dl - I)jt(d) d](p-l)

where r x l denotes the least integer which is �x and jt: Z ---t

{O, ± l} denotes the Moebius function given by

jt(l) = { ( -1 )! : : � :he product of s distinct primes numbers 0, l contains a repeated prime as a factor.

Let a be a primitive element of Fp, and G = {aO,as, . . . , a8(m-l)} a subgroup of F; of order m, where m = (p-l)/s. Next, define two polynomials Jt ('),12(') E F to be equivalent if there exist v E G and 11, E F p such that

fz(x) == Jt(vx) + u,

Clearly this is an equivalence relation. Note that if Jt(x) E F, then Jt(vx) + 11, E F also. Let f(x) E F, and let 11,1,11,2 E Fp, Vl,V2 E G. A little work will show that

'i:z:E G

is possible iff 11,1 = U2 and VI = V2. Thus every equivalence class of polynomials in F contains precisely mp elements. Let FA be the family of polynomials obtained by picking exactly one polynomial from each equivalence class. For instance, this family can be chosen as the set of all polynomials of the form

(5)

where 1 ::; r ::; t and all coefficients Ii are in Fp, but f1' E {cI': 'i = 0,1,···,8 -I}. It follows then that

IFAI = p� ( L (pr(Hl)/d1 -l)jt(d»)

d](p-l)

which simplifies to

IFAI = � ( L p[(t+l)/d1 jt(d)) . p

dl(p-l) .

(6)

We are now ready to construct the arrays that correspond to the CPCW codes in the family. We associate with each polynomial f(.7;) in FA, the (p x m) matrix Af given by

A (i .)={ l, if !(aj)=p - l-i f ,J 0, otherwise

for all i,j, 0::; i::; p -1, j E {a,s, ... ,s(m -I)}. For any two matrices associated with two polynomials f and g (not necessarily distinct), and any integer T

where d is the greatest common divisor (gcd) of the exponents A f (i, j) = Ag (i EBp T, j EBm T) = 1

(�I) of h having nonzero coefficients. Let F denote the set of iff x = aj is a root of the equation all polynomials of the form in (4) that remain, i.e., all those of the form in (4) which are such that the gcd of the exponents f(x) = g(aT x) ElJp T. (7)

450 IEEE TRANSACTIONS ON INFORMATION THEORY. VOL. 41. NO.2.

MARCH 1995

It follows from the manner in which the polynomials in F were selected that (7) has at most t roots except in the trivial case, I = g, unCI T = O.

Let x J(- land XgC) be the sequences associated with the polynomials I and u (via the matrices AI and Ag), respectively. It follows now that if either I # 9 or T # 0

n-l :E x f(k)xg(k EBprn T) <:: t. k=O

Thus the family FA of polynomials yields a CPCW code (we shall also refer to this code as family .FA)' having parameters (n,w,A) = (pm,m,t) and family size IFAI given by (6).

A little work will show that

lim IFAI = 1 rn�oo <I> (pm, m, t)

making the family asymptotically optimal. So the following theorem is proved. Theorem J (Family A): Let p be any prime number,m any

divisor of p - 1, and I any integer, 1 <:: t <:: m - 1. Then the construction described above gives the cyclically permutable constant weight code (pm, m, t) of cardinality M = IF A I given by (6), which is asymptotically optimal (when 'Tn --;. 00 and t is fixed) relative to Johnson upper bound (3),

As mentioned earlier, construction A of this paper overlaps with construction V in [4]. Our approach seems to be more natural and as a consequence our resulting code has cardinality (6) and this number is almost always more than cardinality of the resulting code from paper [4] (see construction V there) with the same parameters.

B. Family B

Let a be an integer 21, p be a prime, and (1 = pa. Let F q denote the finite field of q elements and let t be an integer, I <:: t <:: 1) - l. Note that (p, q - 1) = 1.

Consider in this case, the family of all polynomials over F q of the form

f(x) = Isxs + Is_1X8-1 + . .. + fo where 1 <:: s <:: t and the coefficients fi E Fq for all i. Once again, we would like to discard a subset of these polynomials, namely those polynomials g(:I:) which for some 11 E F: and v E F p satisfy

g(x) = ug(:r + v) ,

A little work along the lines of the previous construction shows that the only polynomials satisfying such a condition (except v = O. 11 = 1) are the constant polynomials

g(:I:) = .rio· Thus we restrict attention to the nonconstant polynomials. Next, declare two nonconstant polynomials h(·). 12(,) to be equivalent if

hex) = Ufl(X + vl, 11 E F;, v E Fp-Clearly, this is an equivalence relation. Note that since we have discarded the constant polynomials, each equivalence

class contains exactly (q - 1)p polynomials. Let FY3 denote the family obtained by picking precisely one polynomial from each equivalence class. Then

qt+l _ q q(qt _ 1) IFBI = = .

(q - 1)[1 p(q - 1)

Next, let f3 be a primitive element in Fq. With any' polyno­mial I in FB, we associate the matrix Af given by

if I(j) = f3q-1-i

otherwise

for all i,j, 0 <:: j <:: p -1, 0 <:: i <:: q - 2. Here a slight modification in the procedure in converting from an array to a codeword belonging to a CPCW code is needed since the arrays are not guaranteed to have constant weight. This is because the column corresponding to a value of :£ for which f(x) = 0 is blank in the array Af. However, since the polynomials in fB can have at most t zeroes, by appropriately replacing l' s with 0' s in the array, we can ensure that the resulting collection of arrays has constant weight p - t.

If we now convert the arrays to sequences, we obtain a CPCW code having parameters (p( q - 1), P - t, t) of size

q( qt _ 1) p(q - 1)

.

Once again from the upper bound (3), it follows that

lim q((/- 1)/p(q - 1) = I p�oo <I>(p(q - 1),p - t, t)

so that this design is also asymptotically optimal. We have proved: Theorem 2 (Family B).' Let p be any prime number and

0', t be natural numbers, (], 2 2, 1 <:: t < P - t. Let q = pa. Then the construction described above gives the cyclically permutable constant weight code (p(q - 1),[1 - t, t) of size M = pa-l (qt - 1) / (q - 1), which is asymptotically optimal (when p --;. ex, and t is fixed) relative to Johnson upper bound (3).

A further generalization of this construction is given in the Appendix.

C. Family C In this final variation of the basic construction, we deal with

rational functions rather than with polynomials. Let q be the power of any prime number 11, p 2 2, Tn any divisor of q - 1 such that m and q + 1 are relatively prime, and t be an integer, 1 <:: t < m/2. Consider the family F of all rational functions

f(x) g(x)

over Fq (Le., both f(·) and g(.) have coefficients in Fq) satisfying the following conditions:

• f and 9 are both nonzero and of degree <:: t. • f and 9 are relatively prime, • I(x)/g(x) "# a, for any a E F: and • f is monic.

MORENO e/ al.: NEW CONSTRUCTIONS OE' OPTIMAL CYCLICALLY PERMUTABLE CONSTANT WEIGHT CODES 451

First we estimate the size of F. The number Id of monic, irreducible polynomials over F q of degree d is given by (see [5, Theorem 4.15], for example)

Id = � L p,(l)l/l. lid

Since every monic. irreducible polynomial over F q of degree d 2 1 divides

we have the simple upper bound qd _ 1

Id < -- d :;, 2.

Of course, h = q. - d '

- (8)

The number of rational functions I (a;) / g( x) in which I and 9 are both nonzero, of degree ::; t, f is monic and f and 9 share a common monic factor h E Fq[x] of degree s, 1 < s ::; t, is given by

(ql-S+1 _ 1 )2 (q - 1)

Let jl(.), jl(·):Fq[x] --> Z, be the function defined by

( -1) I, a( x) is the product of I monic, { L a(x) = 1

jl( a( x)) = irreducible polynomials over F q 0, a(x) contains any repeated

irreducible factor. It follows then that

IFI = L (qt-

s+1 - 1)2 jl(h(x)) - (q _ 1)

hex) (q -1) where the sum is over all monic polynomials hex) E Fq[x] of degree s ::; t, and where the last term accounts for the disallowed constant rational functions.

Using the fact that there is one monic polynomial of degree o and q = h such polynomials of degree 1, we can rewrite this expression as

(qt+1 _ 1)2 IF I = (q -1)

+ degh(x)�2

(qt_l)2 ":"':(-q ---1'--) . q - (q

- 1) ( 1-8+1 1)2 q - jl(h(x)) (q - 1) t (t-.+1 1)2 = t/H1 - q + L q

-L fi,(h(x)).

8=2 (q - 1) dcgh(x)=. Direct calculation for s = 2,3,4,5,6 gives us that

L jl(h(x)) = O. deg h(X)=8

(It seems to us that the above is valid for all s 2 2.) By restricting attention to irreducible hex) (Le., by ignoring double counting), we obtain the simple lower bound

t (t-8+1 1)2 IF I 2 q2Hl - q - L q (q _

�) Is �"I=2

which after some calculation using the upper bound in (8) (for t 2 7), yields

IF I - { 1]2t+l - 1], t = 1,2,3,4,5,6 (9) - 2q2Hl - q2t-6/7, t 2 7.

Denote by c( t) the cardinality of the set F for given t. Now we would like to discard from this set F all those fractions f (x) / g( x) for which there exist u E F q and v E F; such that

f(:c) f(xv) + u

g(x) g(xv)+u'

Clearly, this can happen iff I(x) and f(xv) + u (respectively, g(x) and g(xv) + u) are identical polynomials. In tum, this is possible with v#-1 iff

u = 0 and (d, q - 1) > 1 where d is the gcd of the exponents (2 1) of I(x) (respec­tively, g(x)) having nonzero coefficients. Let Fl denote the set of all fractions from F that remain, i.e., all those in F which are such that the gcd of the exponents corresponding to nonzero coefficients is relatively prime to q - 1.

The number of elements in Fl can be counted with the help of the Moebius function

IFll = L p,( i)c([t/iJ) (10) il(q-l)

where [x] denotes the integral part of x and c(O) = O. Let Ft = Fq U {oo} denote the set of points lying on the

projective line over Ft. This set is of relevance as we will regard the rational functions in Fl as taking on values in Ft. Our next step is to endow Ft with a cyclic structure. Let

1'= [7 1�.q]

be a (2 x 2) matrix over Fq• The elements "" f3 are specified· below.

We identify the vectors

in Fq2 with the element alb E Fq when b #- 0 and with the element x when b = O. With this identification, r may be regarded as a function acting on Ft via

It is easily checked that r is well-defined. Following the approach of BerJekamp and Moreno (see [5,

Theorem 12.11, p. 353]), one can find", and (3 E Fq such that

{ ri [�] I 0 ::; i ::; I] } = Ft·

Moreover

452 IEEE TRANSACTIONS ON IN�URMATION TIlEORY. VOL. 41. NO. 2. MARCH 1995

so that rq+1 is the identity map on F� . Essentially, this is achieved by choosing 1) and f3 such that the eigenvalues of r lie in Fq' and have order (q + 1).

Next, we regard r as a function on Fl by defining

r(f(X») =N( rJ!(:I;) + 13g(x) ) g(x) f(x) + (1 -1])g(x) where, given a rational function, the operator N divides out the common factors between numerator and denominator and in addition, scales the two so as to make the numerator monic. More precisely, given 71(:1:)/,11(:1:) E Fq(:E),

lV(U(X») = u'(x) vex) v' (x)

in which u'(x) and v'(x) are relatively prime, u'(x) is monic and

u(x) u'(x) vex) v'(x)'

Clearly, even this map is well-defined. We claim that, given

f(x)/g(x) E Fl r(f(x») = 71(:1:) E Fl g(x) vex)

as well. Clearly, u and v are nonzero polynomials of degree �t that are relatively prime and u is monic. It remains to verify that u(x)/v(x) t= a, for some a E F�. But this is not hard to show, using the fact that f(:I:) and g(:E) are relatively prime.

In our code construction, we will need rational functions

f(x)/g(x) which are such that

;�;� t= rk G�:;D for any integer k, 0 � k � q and any element a, a E F� (a t= 1. if k = 0). It turns out that every rational function in Fl satisfies this requirement. To see this, note first of all that

rG�;Dlx=u = l'G��D· Next, assume that there exists f(x)/g(x) E Fl for which

f(x) =

rk(f(ax») g(x) g(ax) for some integer k, 0 � k � q, and some element a, a E F: (a t= 1, if k = 0). However, setting x = 0 in the above, gives us a contradiction since r cycles through the elements in F+ and has period q + 1.

q

Let G be some subgroup of order m of multiplicative group F: and 8 a generating element of C. Next, define two functions h(X)/gl(X) and h(X)/g2(X) to be equivalent if

hex) = rk (h(ax») g1 (x) g2(ax) for some integer k, 0 � k � q, and some element a, a E G. Clearly, this partitions the set Fl and the number of equivalence classes equals

IFll (q+l)m'

Let Fe denote the set obtained by picking precisely one element from each equivalence class. We associate with the element

in Fe, the (q + 1) x 771 matrix Ak given by

Ak(q + 1 -i,j) =

{ l. if for someaEF*. [afk(Oi)] =ri[l] x ' .

q agk(fP) 0 0, otherwise.

(11)

Clearly, by our choice of the subset Fe, we have ensured that the two-dimensional periodic auto- and cross-correlation of the matrices {Ad is bounded above by 2t (the maximum number of common zeroes of two distinct rational functions with both numerator and denominator nonzero and of degree bounded above by t). Thus this construction leads to a « q + l)m, 771, 2t) cyclically permutable constant weight code of size

IFel = IFll/m(q + 1), given by (9) and (10). Once again it follows from the upper bound on <p( n, w, A)

in (3) that the design is asymptotically optimum. This gives us Theorem 3 (Family C): Let q = p', where 8 :;:. 1, p is any

prime number, 771 is any divisor of q - 1 such that m and q + 1 are relatively prime, and t is any integer, 1 � t < m/2. Then the construction described above gives the cyclically permutable constant weight code « q + l)m, m, 2t) of size M = IFel = IF11/m(q+1), defined by (9) and (10), which is asymptotically optimal (when m --; 00 and t is fixed) relative to Johnson upper bound (3).

III. FURTIlER GENERALIZATIONS

All three constructions A, B, C may be rewritten in another way, similar to the standard concatenation construction [6J. From this point of view [4, Theorem 1] is of interest to us. In this theorem any p-ary linear cyclic (n, k, d) code is used (as outer code) and a binary vector v of length p and weight w(v) (as inner code). When w(v) = 1 and the (n,k,d) code is MDS, then, as it was shown in [6], the resulting constant weight code is asymptotically (when p --; 00) optimal relative to the Johnson upper bound (3). When w(v) > 1, the bound (3) is no longer valid for the resulting code. Here we extend these constructions with weights of inner codes w(v) > 1. When the parameters correspond, our constructions often give better codes.

Let us have a CPCW code B with parameters (nb' Wb, Ab) of size Mb. Going back to the definition of Ab (see (1) and (2), we may write Ab = max (Aa, Ac), where Aa is the maximal autocorrelation (1) and Ae is the maximal cross-correlation (2). Denote as code B the one with parameters nb , Wb, Aa, Ac and Mb as a code B:(nb,wb,Aa,Ae,Mb)'

Theorem 4 (Family Ae): Let p be any prime number, 771 be any divisor of p - I,m = (p - 1)/8, t be any integer, 1 � t < m, a be a primitive element of Fp, and FA be the

MORENO et al.: NEW CONSTRUCTIONS OF OPTIMAL CYCLICALLY PERMUTABLE CONSTANT WEIGHT CODES 453

set of all polynomials over:Fp of form (5). Let B be a CPCW code (p, Wb, Aa, Ac, Mb). Then the set C of all vectors Xf,b

where x f,b corresponds to the polynomial j and to the vector b = (bo, bl,"', bp-1) via the Chinese remainder theorem and .the matrix A f,b of size p x m,

Af,b(i,j) = bie'lpU, where u = j(aS}) (12)

is the CPCW code (n,w,A) of size M, where

n=mp, W=mwb, A=max{mAc,(m-t)Aa+twb} M = I:FAI· Mb (13)

where I:FAI defined by (6). Proof The length n of the resulting code C follows from

the dimension of the matrix Af,b. The value

u = .f( aSj) E {O, 1, ... ,p - I} = F p

does not depend on i, so, when i goes over all elements 0,1,·· . ,p -1, so does the value i EElp u. This gives the weight w of vector x f,b. Now we want to find the correlation between any two distinct vectors :I: j.b and Yg,c' Define two vectors :I: and Y of length mover F p'

x = (xo, . . . , Xm-l), Xi = j( a8i) (14)

Y = (Yo, . .. , Ym-l), Yi = g(a8i). (15)

Definc thc operator u that cyclically moves components of the vector b to the right: if b = (bo,bl,··· ,bp_I), then ub = (bp-1,bo,bl,···,bp-2). Let (Jill denote u(ui-1b) for i = 1" " , p. With this notation the matrices A j,b and Ag,c can be rewritten as follows:

A j,b = [uxo (bf, UXt (b)T, ... , uX=-t (bfJ (16)

Ag,c = [uYO(cf,uY1(c)T, ... ,uY=-l(b)Tj (17)

where "TOO means transposition. We have to consider the following cases.

The case b =F c. By the conditions of the theorem the vectors b, c belong to the CPCW code B: (p, Wb, Aa, Ae, Mb). So for any i E Fp the correlation A(uib, c) � Ac and therefore the correlation

A(Xj,b,Yg,c) � mAc. The case b = c. If f =F g we know from Theorem 1 that

vectors x and y, defined by (14) and (15), coincide in not more than in t positions. So

If f = g, but X j,b =F Yg,c, it means that they differ by cyclic shifts, Cyclic shifts of vectors x j.b and Yg,c mean cyclic shifts of vectors x and Y and (or) cyclic shifts of vectors band c. Let :1; and y differ by cyclic shifts (that is, ukx = Y for some k, 1 � k � m - 1). Then, as we know from construction A, uk x and x coincide not more than in t positions. Thus we obtained for this case

This gives the expression for A in Theorem 4. Now we can conclude also that the size M of code C is equal to the product of I:F A I and Mb, and this finishes the proof of the theorem.

Example 1: As we know (see [5, ch. 15]), the (cyclic) codes which are dual to binary BCH codes of length 28 -1 correcting two errors, have distance 28-1_2(8-1)/2 and (2S _ 1)(28-1+ 1) codewords of weight 2,,-1 (8 is odd). It gives us a CPCW code with the following parameters:

nb = 28 -1, W = (nb + 1)/2, Aa = (nb + 1)/4 1 Ac = (nb + 1)/4 + 2v(nb + 1)/2, Mb = (nb + 3)/2.

When nb = P is a (Mersenne) prime number, we can apply our construction A. e and obtain the family of CPCW codes with the following parameters (take here m = p - 1):

2 p+3 n = pep - 1), W = (p -1)/2, M = -2- I:FAI

A = max {(p _ 1) (p: 1 + �jP; 1) ,

P+l} (p+T-l)'-4 -

where I:FAI is defined by (6). When T 2: (p-l) / VP + 1 it gives the cyclically permutable

constant weight codes with the same parameters (n, w, A) as in [41 (see [4, construction III)), but with (p + 3)/2 times more codewords.

A similar generalization may be obtained for construction C. Let us assume that we are under the conditions of Theorem 3. That is, q = pS, where 8 2: 1 and p is a prime number. Let m be any divisor of q - 1, G be a subgroup of F; of order m, and (J be a generator of G, G = {(J0, (JI," . ,(Jm-l}. Suppose that we have a CPCW code B with parameters (q + l,wb,Aa,Ae,Mb)

B = {b = (bo,bl,'" ,bq)}

and let :Fe be the set of fractions defined in p, 2.3:

:Fe = {Rk(X) = ::i:�: k = 1,2"", 1:F1 1/m(q + I)}, (18)

For any fraction Rk(X) E :Fe and any vector b E B define the matrix Ak,b of size (q + 1) x m

Ak,b(i,j) = biEb,+tU where u is such that for some a E F;

[��:��;n = rq+l-u [�l (19)

Theorem 5 (Family ce): Let q = p', where 8 2: 1 and p is a prime. Let m be any divisor of q - 1, such that m and q + 1 are respectively prime, G be a subgroup of F; of order m, and let G = {(Jo,BI, . . . ,(Jm-I}. Let:Fe be the set of fractions (18), defined on p. 2.3, and assume that we have a CPCW code B with parameters (q+ l,wb,Aa,Ae,Mh). Then the set C of all vectors Xk,b

C = {Xk,b:Rk(X) E Fc,b E B}

454 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO.2, MARCH 1995

Fam.

A

B

c

TABLE I PARAMETERS OF THE THREE CONSTRUCTIONS

1'"' pprime

mllp-I)

(q - l)p q = p�

p prime a 2:: 1

m(q+ 1) q = pc. p prime a� 1

ml(q-I) (m,q+l)�1

(p-I)

1 :St::;m

I 1';; I';; (p-I)

21 1S;tS;m/2

Size

...!... 2: prc'+')/�"(d) mp,j)(,._l)

! (t.=.!.) P q-l

_1_ 2: ,([I/dJl, whe .. (q+l)m4)(q_l)

C,,)_ { q�t+l_q. 1�t�6 l - � (i"+l _ q21�/7, t?: 7

where Xk,b corresponds to the fraction Rk(X) and the vector b through Chinese remainder theorem and matrix Ak,b of form (19), is CPCW code (n,w,.>..) of size M, where

n = m(q + 1), w = mWb

.>.. = max(rn'>"e, (m -2t)'>"a + 2twb)

M = Mbl.Fcl where l.Fel defined by (9), (10), and (18).

The proof is similar to the proof of Theorem 4, Example 2: Let p = 7 and m = 6, U:t B: (7,3,1,2,2)

consist of' the two vectors

B = {(1l01000), (lOnOOO)}.

Using the construction Ae with t = 3, we obtain a CPCW code with parameters n = 42, W = 18,'>" = 12 (that is, d = 12), and M = I.FAINh = 110. For the same p = 7 and m = 6 construction A with t = 2 gives a CPCW code with parameters n = 42,w = 6,'>" = 2, and 1'\lJ. = I.FAI = 7.

ApPENDIX

i) In the alternative construction and its generalization, we construct first a none on stant weight code, then we make it a constant weight one by deleting some 1 's from the codewords. Another way to make the code a constant weight one is to delete all codewords with weights smaller than p from the code. This is equivalent to using a subset of if (or a subset of if,r) to construct the code. This subset is

:tin = {J E :tr! has no root in GF(p)}

or

:t:,; = {J E n,r:! has no root in GF(p)}.

Let the cardinality of these sets be denoted by Nt{p) and Nt{p, r), respectively, then

Theorem: Nt (p, r) satisfies the following equation:

t (p + s _ 1) rt+l - r' L Nt-s(p, r) = ( ) '

8=0 S P r - 1

Proo!' Let Mg(p) be the number of monic polynomials of degree s having exactly s roots in GF(p). We have

M8{p) = (p + : - 1) .

Let N., (p, r) be the number of polynomials with coefficients in GFCr) of degree s having no root in GF(p). Then the set of polynomials of degree s :s t can be partitioned into subsets of polynomials with exactly i roots in GF(p), i = 0,2,···, .s. The form of the polynomials with exactly 'i roots in GF(p) can be written in the form

!(X) = II (x - lj)g(x) j=1

where g(x) is a polynomial of degree s - i having no root in GF(p) and Ij E GF(p) for all j. Therefore, the number of these kinds of polynomials is (p + i-I) A

i Nt-s(p, r). This results in

� (p + S - 1) , t+1 � s Nt-s(p,r) = r - r .

The right-hand side is the number of all polynomials of degree at most t which is not a constant. If we consider only the polynomials in the set :ti'T, we can see

N ( ) _ il.,(p, r) s p,r - p(r-l)' This leads to the conclusion of the theorem.

This code has parameters (p(r - 1), p , t) and cardinality Nt(p , r) . As a consequence of the theorem, we have

lim lim Nt{p) = e-pU-1 t�oop�oo <I>(p(T - 1), p , t)

where T = pa. Since Nt(p) = Nt(p,p), the variation for the code in (i) is also covered by this theorem.

ii) As a generalization of this variation, we may use the following subset of if'T:

:t:,�. = {! E :t;,T:! has at most s roots in GF(p)}. By using (his subset, we obtain a code with codeword

weights from p - s to p. By deleting some I's from the codewords of weights greater than p -s, we obtain a constant weight code of weight p - s. The parameter of this code is (p(T - 1),p - s,t). We can see that the cardinality of the code is

� (p+ j -1) M(t,s,p) = � . Nt_j(p,T). j=O J

These families of codes are no longer asymptotically opti­mal. Actually, we have

I· I' M(t, s , p) 1m 1m t�ocp�co <I>(p(T -1),p - 5, t)

s

exp (p-a+l) - L []I-(a-l)j]/ j! j=1

MORENO e/ ai.: NEW CONSlRUCTIONS OF OPTIMAL CYCLICALLY PERMUTABLE CONSTANT WEIGHT CODES 455

REFERENCES

[I] F. R. K. Chung, J. A. Salehi, and V. K. Wei, "Optical orthogonal codes: design, analysis and applications." [EEE Tran.'. Inform. Theory. vol. 35, pp. 595-604, May 1989.

[2] E. F. Brickell and V. K. Wei, "Optical orthogonal codes and cyclic block designs," Congressus Numeranlium, vol. 58, pp. 175-192, 1987.

[3] H. Chung and P. V. Kumar, "Optical orthogonal codes-New bounds and an optimal construction." [EEE Trans. Inform. Theory, vol. 36, no. 4, pp. 866-873, July 1990.

[4] Q. A. Nguyen, L. Gyorfi. and J. L. Massey, "Con<lructions of binary constant-weight cyclic codes and cyclically permutable codes," [EEE Trans. [nform. Theory, vol. 38, no. 3, pp. 940--949, May 1992.

[5] F. J. McWilliams and N. 1. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977.

[6] V. A. Zinoviev, "Cascade equal weight codes and maximal packings," Prob!. Contr. Inform. Theory, vol. 12, no. I, pp. 3-10, 1983.

l7J A. E. Brouwer, J. B. Shearer, N. J. A. Sloane, and W. D. Smith, "A new table of constant weight codes," IEEE Trans. Inform. Theory. vol. 36, no. 6, pp. 1334-1380, Nov. 1990.