~

678 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991

rate E I 1/2 was studied. For odd values of m , P J E ) satisfies the 2-p bound. For even values of m , this is not necessarily true. However, for m 2 10, the 2-” bound is almost satisfied. The extended codes were found to possess similar characteris- tics.

APPENDIX

Lemma 1: Let f (E) be a continuously differentiable function defined on [0,1/2] and satisfying f(0) 2 0. If

f ’ ( ~ ) 2 - a ( E ) f ( E ) , 0 I E I 1/2, for some nonnegative function ( Y ( E ) , then

f ( € ) 2 0, 0 s E I 1 / 2 .

Proof: See [2]. 0

Lemma 2: For 1 = 4, 8, 16; . . , g / ( E ) A (102016-226814+120012 -180)~~+(11401’ - 7 8 0 ) ~ ~

- ( 5 3 0 Z 2 - 3 3 0 ) ~ 2 - ( 6 0 1 2 - 6 0 ) ~ + 1 5 1 2 - 1 5 ~ 0 . (26)

Proof: We will use mathematical induction for the proof. At E = 0, g4(e) is positive. Moreover g4(e) has no real roots. Hence g 4 ( E ) 2 0 . We now assume that g I ( E ) 2 0 and prove that gZI(E) 2 0. From (26) we can write

g Z I ( E ) = (6528016 -3628814 +480012 - 1 0 8 ) ~ ~

+(4560l2 - 7 8 0 ) ~ ~ -(204Ol2 -330)~‘ -(2401’ - 6 0 ) ~

+6012 - 15 2 0. (27) We can express g Z I ( e ) as

where

h ~ ( ~ ) ~ ( 4 2 8 4 1 ~ - 2 2 6 8 1 ’ + 2 4 0 ) ~ ~ + 2 2 8 ~ ~ - 1 0 2 ~ ~ - 1 2 ~ + 3 .

(29)

Since g I ( E ) 2 0 by assumption, it is sufficient to show that h / ( ~ ) 2 0. For this we again use mathematical induction. At E = 0, h4(e ) is positive. Moreover, h4(c) has no real roots. Hence h 4 ( e ) 2 0 . We now assume that hl (e)2O and prove that h 2 [ ( ~ ) 2 0. From (29) we have

h21(~)=(6854414-907212 + 2 4 0 ) ~ ~ + 2 2 8 ~ ~ - 1 0 2 ~ ~ - 1 2 ~ + 3

(30)

= h/(E) + T I ( € ) , (31)

T I ( € ) = (6426014 -68O4l2)e4. (32)

hZI(E) 2 0. 0

where

Since h l ( e ) 2 O by assumption and rI (e)>O, it follows that

REFERENCES

[ l ] C. Leung and M. E. Hellman, “Concerning a bound on undetected error probability,” IEEE Tmns. Inform. Theory, vol. IT-22, no. 2, pp. 235-237, Mar. 1976.

[2] C. Leung, E. R. Barnes, and D. U. Friedman, “On some properties of the undetected error probability of linear codes,” IBM Res. Rep., RC 7194 (#30940), June 1978. Portions of this report were published in IEEE Truns. Inform. Theory, vol. IT-25, no. 1, pp. 110-112, Jan. 1979.

[3] J. K. Wolf, A. M. Michelson, and A. H. Levesque, “On the probabil- ity of undetected error for linear block codes,” IEEE Trans. Com- mun., vol. COM-30, pp. 317-324, Feb. 1982.

[4] T. Kasami, T. Move, and S. Lin, “Linear block codes for error detection,” IEEE Trans. Inform. Theory, vol. IT-29, no. 1, pp. 131-136, Jan. 1983.

[SI T. Kasami and S. Lin, “On the probability of undetected error for the maximum distance separable codes,” IEEE Trans. Commun., vol. COM-32, pp. 998-1006, Sept. 1984.

[61 S. Lin and D. J. Costello, Error Control Coding: Fundamentals and Applications.

[7] F. J. MacWilliams, “A theorem on the distribution of weights in a systematic code,” Bell S y s f . Tech. J . , vol. 42, pp. 79-94, Jan. 1963.

[SI S . M. Selby, Standard Mathematical Tables, 21st ed. Cleveland, OH: CRC Press, 1973, p. 106.

Englewood Cliffs, NJ: Prentice-Hall, 1983.

A Note on Perfect Multiple Coverings of the Hamming Space

G. J. M. van Wee, G. D. Cohen, and S. N. Litsyn

Abstrucf-kt Q be an alphabet of size q 2 2. The Hamming space Q‘ that consists of all n-tuples of elements of Q is a metric space, provided with the Hamming distance function. A perfect multiple cover- ing (PMC) is a code C in Q” such that there exist fixed numbers r and p with the property that every word in Qfl is within distance r from exactly p codewords of C. We give a few constructions of PMC’s, and investigate in detail the problem of determining all possible parameters of PMC’s with r = 1.

Index Terms -Multiple coverings, covering radius, perfect codes, foot- ball pool problems.

I. INTRODUCTION

Let q and n be integers, q 2 2 and n k l . Let Q be an alphabet of size q. In this correspondence we usually take Q equal to Z,, the group of integers modulo q. Sometimes, if q is a prime power, we take Q = F,, the finite field with q elements. The set of n-tuples of elements of Q is a metric space, with the (Hamming-) distance d defined by d b , y) = I{ilx, # y,)l, where x = ( x , , x 2 ; . ., x , ) , y = ( y l , y 2 ; . ., y , ) E Q f l . Sometimes it is more convenient to write x l x z . . . x , instead of ( x , , x 2 ; . ., x,). The elements of Q f l are usually called words. We also define wt(x) = d(0, x), the weight of x, where the word with all coordi- nates equal to zero is denoted by 0.

Let K be a set of possible messages, and suppose that M := IK( I q”. Encoding the set of messages K means that one chooses an injective map 4: K -+ Q,. The subset C := $ ( K ) of Qn is called a code (hence any nonempty subset of Q f l is a code). If q is a prime power, and C is a linear subspace of E:, C is called a linear code over E,. When we want to send a message

Manuscript received July 12, 1990. This work was supported in part by the Netherlands Organization for Scientific Research ( W O ) . This work was presented at “Designs and Codes,” Obewolfach, Germany, April

G . J. M. van Wee is with the Eindhoven University of Technology, Department of Mathematics and Computer Science, P.O. Box 513,5600 MB Eindhoven, The Netherlands.

G. D. Cohen is with Ecole Nationale SupCrieure des TCICcommunica- tions, 46 rue Barrault, 75634 Paris Cedex 13, France.

S. N. Litsyn is with Perm Agricultural Institute, Department of Cybernetics, Kommunisticheskaya, 23a, 614000, Perm, U.S.S.R.

IEEE Log Number 9041810.

1-7, 1990.

0018-9448/91/0500-0678$01.00 01991 IEEE

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991 679

k E K over a channel, the 4-ary symmetric memoryless channel say (see [8, p. ll]), the associated word x := 4 ( k ) E Q" is trans- mitted over this channel. The receiver receives a word y , which might differ from x, because of noise on the channel. The word y need not even be in the code C . In general we have y = x + e, where e E Q" is the error vector. The weight of e is the number of errors that occurred during the transmission of x.

Nearest neighbor decoding (cf. [8, p. 91) means that the re- ceiver determines an x' E C , such that d ( y , x') = d ( y , C ) . The outcome of this decoding procedure is this x', or, in fact, the unique message k' such that 4 ( k ' ) = x'. We will not make this distinction between codewords and messages anymore. Correct decoding in this sense means that x = x'.

Under certain circumstances a different decoding procedure, called list decoding, can be useful. We shall not go into details about the practical purpose of list decoding. For this, see [l] and [3]. We only describe the procedure here. Let p be a positive integer. When y is received, in the list decoding procedure the p codewords in C closest to y are determined. More precisely, a set (list) L of p codewords in C is determined, such that for all X ' E L and all c E C\L, d ( x ' , y ) I d ( c , y ) . The list L is the output of this decoding procedure. Correct decoding means that the originally transmitted word x is in the list L. The nearest neighbor decoding procedure, described earlier, is a special instance of list decoding where we choose the parameter p equal to 1.

For x E Q", the set { y E Q"ld(x, y ) I r } is called the sphere with radius r and center x, and it is denoted by B,(x). We put

V , ( n , r ) := IB,(x)I = ( 7 ) ( 4 - i = O

Now let p be fixed. Suppose we want to know how many errors our list decoder can handle. That is, we ask for the maximal number r , with r I n, such that the following holds: assuming that we always have wt(e) I r , the list decoder decodes cor- rectly, for any x E C that is transmitted. One easily verifies that this number equals

max{ r E (O,l,. . . , n } Wx E Q" :I B,( x) n C I I p } . This brings us closer to the actual subject of this correspon- dence. A code C c Q" with

I B r ( x ) n c l i p , fo ra l lxEQ" (1) is called a p-fold r-packing of Q". We have IClV,(n, r ) I pq". A code C with

I B , ( x ) n C [ > p , forall X E Q "

is called a p-fold r-couering of Q". In this case, IClV,(n, r ) 2 p q " . If a code C satisfies both (1) and (21, it is called perfect. We then have

(3) (the sphere packing or covering equation). One often speaks of (perfect) multiple packings or coverings.

A ( 4 , n , M , r , p ) perfect multiple covering (PMC, cf. [2]) de- notes a p-fold r-covering of Q", with lQl=4, which has M codewords. For example, the code C that equals Q" is a (q , n , q", r , V,(n, r ) ) PMC for any r E (0,l; . . , n}. Other trivial examples are the codes with r = n: every code in Q" is a ( 4 , n , M , n , M ) PMC, where M is the cardinality of the code.

In this correspondence we investigate for which parameters PMC's exist. We shall pay most attention to the case r = 1. In Section I1 we give a few simple constructions of PMC's. In

Section I11 we determine all possible parameters of PMC's in the case r = 1 (Theorems 2 and 3), provided q is a prime. This restriction on 4 can be dropped if we restrict ourselves to linear PMC's. We also give a result that applies in the case that none of these two conditions is fulfilled (Theorem 4 and Corollary 1). After reading this correspondence, it will be clear that in this area there is still a lot of work left to be done. One of the goals of this correspondence, is to stimulate further research.

When p = 1 ( r arbitrary), we are simply dealing with perfect codes in the ordinary sense (see [8, p. 191). The determination of all parameters 4 , n , r for which such a code exists is a well-known and widely studied problem. Extensive research has been done on this. For a survey and many references, see [5].

Multiple packings and coverings, perfect or not, are the topic of Clayton's thesis [2]. This thesis contains many interesting results. In [2], the definition of a PMC is somewhat different from the definition previously given: Clayton allows a "code" to be a multiset, i.e., a code can have repeated codewords. Every PMC by our definition is a PMC by Clayton's definition, but not the other way around. In this context, [2, Corollary 4.71 is most interesting. It says that if the parameters 4 , n , r admit a PMC (repeated codewords allowed), then there also exists a subgroup of Zi, which is a ( 4 , n , q"-', r , V,(n, r ) / q ) PMC (without re- peated codewords). Therefore, the results in [2] concerning the (non-) existence of PMC's also apply if we use our definition. For the sake of clearness, we only use our own definition in this paper.

Related to his different definition of a PMC, is the fact that Clayton always poses the question as follows: Given q, n , and r , does a ( q , n , M , r , p ) PMC for some p and some M with 1 < M < 4" exist? In other words, the parameter p is not fixed ( M follows from the parameters q, n , r , p by (3)). In this corre- spondence we investigate a different problem: For which 4 , n , r , p , does a ( 4 , n , ., r , p) PMC exist?

From the numerous results in [2], we only mention two (see [2, Theorems 4.8 and 4.91).

For given r and q , there always exists a ( 4 , n , M , r , p ) PMC for some n, p, and M with r < n and M < 4 " . More precisely, for given r and q , there exists a PMC for n = r4

(4) For r = 1, a PMC exists for some p and M with M < 4" if

(5 )

+ 1, M = qr4 = q"-', and p = V,(n, r ) / q .

and only if n = 1 (mod 4).

In this introduction we mentioned list decoding in connection to (perfect) multiple packings and coverings. Multiple coverings also come up in a different natural way. Suppose we have a football pool with 7 matches, each with 4 = 3 possible outcomes: win, lose, or draw. A forecast wins a prize if for at most one of the matches the predicted outcome is wrong. After the matches have been played, there is usually more than one winning forecast, and in this case the money has to be split. Now if a participant has p winning forecasts by himself, and the total number of winning forecasts in the pool is much greater than p, then he earns about p times as much money as when he had only one winning forecast.

Suppose a participant wants to be sure that he wins a prize, whatever the outcomes of the matches are. Then he needs a 1-fold 1-covering of IF:. The smallest such covering known at present has 186 codewords, and is given in [4]. If this participant wants to win p = 5 prizes, he could of course send in five times these 186 forecasts. This would take 5.186 = 930 forecasts. This strategy is not profitable since, for five times the costs, he earns

680 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991

(less than) five times as much money. However, he can reach his goal in a cheaper way, namely by using a (3,7,36 = 729,1,5) PMC, which exists by Theorem 2a). In this way the costs are considerably less than five times as high! By [7, Table I], any 1-fold 1-covering of 53 has at least 146 codewords, so even when the number 186 of [4] is improved, the first method still needs at least 5.146 = 730 > 729 forecasts.

11. SOME CONSTRUCTIONS

In this section we mention a few simple ways to construct PMC's. Constructions l), 2), and 3) were already mentioned in the Introduction.

Every ordinary perfect code is a PMC with p = 1. Exam- ples are binary repetition codes of odd length, Hamming codes, binary Golay code, ternary Golay code (see 18, p. 179, 1801). The PMC's with r = n : every code C in Q" is a ( q , n , M , n , M ) PMC, where IC1 = M . C = Q " ; this is a (q ,n ,q" , r ,V , (n , r ) ) PMC for any r E {O, 1, * . , n}. "Complementing": This only works for binary codes. From any (2,n, M , r , p ) PMC C with r < n, we obtain a (2, n, M , n - r - 1, M - p ) PMC, by replacing every code- word of C by its complement (that is, replace all zeros by ones and all ones by zeros). "shifting" of a group code: Suppose a (q , n, M, r , p ) PMC C is a subgroup of Q". For any p'~{1,2;. . ,q"/M), we get a (q , n, p'M, r , p'p) PMC C' if we take C' equal to the union of C and p'- 1 of its cosets. In particular, this works if q is a prime power, Q = 5, and C is a linear PMC.

6) The "Vasil'ev construction" (see [6, p. 901 and 18, p. 771): Suppose q is a prime power and let F , = { O = a , , (Y,;~~,(Y,-~). Let C be a code in F:. Define the maps T: E?-')" -+ F, and T : F?-I)" + F: by

q - 1 n

; = I j = l ~ ( v ) = C ai C ~ ( i - I ) n + j )

and

T ( v ) = (TI( U ) 9 T,( U),. . ., T"(U)) 7

where q - l

?(U)= C ( I I j s n ) , i = l

and

U = (UI,U2,...,U(q-,)n) E F y ) " .

Define the code C'gF;"+' by

C':= { (v lc + T ( v ) l p ( v ) ) E F:fl+lI~ E F2-l)" and c E C } .

We have IC'J = q(Q-')"1C[. We summarize a few important prop- erties of this construction in Theorem 1.

Theorem 1: Construction 6) has the following properties:

a) If C is a p-fold 1-packing, then so is C'. b) Let r E {1,2; . 1 , n}. If C is a p-fold r-covering, then so

c) If C is a (q ,n ,M, l ,p) PMC, then C' is a (q ,qn+ l ,

d) If C is linear, then so is C'.

is C'.

q(q-')"M, 1, p ) PMC.

Remark 1: Theorem 1 a) does not generalize to p-fold r-pack- ings with r > 1. For instance, if C is a 1-fold 2-packing with q = 2, n = 3, and [Cl = 1, then C' is not a 1-fold 2-packing by the sphere-packing bound.

Proof of Theorem 1: a) Suppose an x E E:"+' and p + 1 different codewords

b('), b('); . ., b(''+l) E C' exist such that for k = 1, 2; . ., ~ + l , d ( x , b ( ~ ) ) 1 1 . Write

x = ( y l z l a ) , with y E Fp-l)", z E F: and (Y E F,. For k = 1,2,. . ., p + 1, write

b ( k ) := ( u ( ~ ) ~ c ( ~ ) + T ( v ( k ) ) l ~ ( , ( k ) ) ) ,

where d k ) E E?-')" and d k ) E C. We claim that 1) d(z,&" + T ( y ) ) I 1, for k = 1,2; . ' , p + 1, and 2) the d k ) are all different. This gives a contradiction with IB,(z - T ( y ) ) n C1 I p. Hence, IB,(x)n C'I I p for all x E Fy+ ' , i.e., C' is a p-fold 1-packing.

First we prove 1). Let k E {1,2,. . . , p + 1). Put d , := d ( y , d k ) ) , d , := d ( z , d k ) + T ( d k ) ) ) and d ; := d ( z , d k ) + T ( y ) ) . From d ( x , b ( k ) ) I 1, it follows that d, I 1, d , I 1, and d , + d , I 1. If d , = 0, then T ( y ) = T(dk) ) , hence d ; = d , I 1. If d , = 1, then d , = 0 and d ( T ( y ) , T ( d k ) ) ) = 1. Hence, by the triangle inequal- ity, d ; I d , + 1 = O + 1 = 1. So in both cases d ; I 1, which proves 1).

To prove 2), suppose that we had i , j E {1,2; . . , p + 1) with i # j and di) = d'). Put

d := d(b ( ' ) , b ( I ) ) , d , := d(U( ' ) , v ( j ) ) ,

d , := d ( c(') + T ( U(')), c ( j ) + T ( U(')))

= d ( T ( d ' ) ) , T ( d ' ) ) ) ,

and

d , := d ( T( U(')), T( U''))).

We have

O < d l + d , + d , = d ~ d ( x , b ( ' ) ) + d ( x , b ( ' ) ) ~ 1 + 1 = 2 .

If d , = O , then

T ( U(')) = T( U(')) and n-( U(')) = T( U(')),

hence d , = d , = 0, contradicting d > 0. If d , = 1, then

d( T ( d ' ) ) , T ( d ' ) ) ) = 1 and "(U(')) # ~ ( d ' ) ) , hence d , = d , = 1, contradicting d I 2. Finally, if d , = 2, it follows from d I 2 that d , = d ,=O. Let ( e - l ) n + g and ( f - l)n + h be the two coordinates in which U(') differs from U('), where e , f E {1,2; . ., q - 1) and g, h E {1,2; . ., n). Define D, Y E E,\{O) by

U{:)- 1)" + g - U{:' I),, + g = p and

From d , = 0 it follows that g = h and y = - p. From d, = 0 and y = - p it follows that (Y$ = - afy = afP, which implies e = f . Consequently, ( e - 1)n + g = ( f - 1)n + h, a contradiction. This completes the proof of 2), and thereby the proof of Part a) of the theorem.

b) Let x E E r + ' . Write x = (y l z la) , with y E FP-I)", z E F: and (Y E IF,. There are p different codewords dl), d2), . . . ,e('') E C , such that d ( z - T ( y ) , d k ) ) < r ( k =1,2;..,p). We shall now define p different (!) words b ( k ) E F y + l that are all in the code C', and that satisfy d ( x , b ( k ) ) I r ( k = 1,2; . ., p). Since x was arbitrary, this proves that C' is a p-fold r-covering. We distin- guish between two cases: 1) and 2).

U{;: 1)" + h - U { { y I)" + h = ?'.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991

___

68 1

Case 1) a = ~ ( y ) . Then, for k = 1,2;. . , p , define

b'k' := ( y l P + T( y)I7r( y)) . Case 2) a # ~ ( y ) . For every k such that d ( z - T ( y ) , d k ) ) I r

- 1, define b(k) as in case 1). For the other k , we do the following. We have d( z - T ( y ) , d k ) ) = r 2 1. Let pk ~ ( 1 , 2 ; . . , n ) be a coordinate in which z - T ( y ) and d k ) differ. Let s(k)E{1,2;. * , 4 - l} be such, that the pkth coordinate of Z- T ( y ) - c ( ~ ) is a,(k). Define t ( k ) E (1,2; . e, 4 - 11 by at(k)as(k) = a - ~ ( y ) . Let w be the word in 59-l)" with the ( ( t ( k ) - 1)n + pk)th coordinate equal to a,(k) and all other coordinates zero. We have d(z , d k ) + T ( y + w)) =

r - 1 and ~ ( y + w ) = a. Now define

:= ( y + wIdk) + T ( y + w ) l a ) .

c) This follows immediately from a) and b). A different way to prove it is by combining a) (orb)) and the fact that C' satisfies the sphere packing equation (31, because C does:

IC'IV,(qn+1,1)=4(q-1)flM(1+(4n+l)(q-1))

= @ - 1)" + 1M( 1 + 4 4 - 1)) = 4 ( 4 - l ) n + l p 4 n = p 4 4 n + l .

d) Trivial. 0

Example: The code C = {00,01, 10,111 = 52' is a linear (2,2,4,1,3) PMC, (cf. construction 3)). For this C, construction 6) gives

C' = (00o0o,00010,00100,00110,

01011,01001,01111,01101, 10101,10111,10001,10011, 11110,11100,11010,11000},

a linear (2,5,16,1,3) PMC.

111. PERFECT MULTIPLE COVERINGS

In this section we discuss the case r = 1. We try to determine all possible parameters 4, n, M, p for which a ( 4 , n, M, 1, p ) PMC exists. Of course, it is sufficient only to specify 4 , n, and p, since M follows from the other parameters by (3).

Let us first return to the ordinary case, p = 1 . Let 4 be a prime power and let C, be the code {O} in 5:. This is a (4,1,1,1,1) PMC. We apply construction 6) of the previous section to C,, and obtain a linear ( 4 , 4 + 1, q',- '), 1, l ) PMC C, in IF4+' . If we apply Construction 6) to C,, we get a linear

infinitely. We get an infinite sequence of linear PMC's {Cir;"= 1,

where Ci has the parameters ( 4 , ( 4 ' - l ) / ( q - l), q(+l)/(q-l)-i ,1,1). This sequence of codes is exactly the well- known sequence of Hamming codes over 5, (see [8, p. 1931). In Theorem 2 we generalize this, and construct sequences of per- fect multiple 1-coverings for any p.

It is also well known (see [SI or [8, p. 1801, for example), that, for 4 a prime power, the parameters of the Hamming codes are the only parameters for which there exists a (linear or nonlinear) perfect 1-error-correcting code over 5,, that is, a ( 4 , n, M, 1,l) PMC. We shall show that a similar statement is true in the case of perfect multiple 1-coverings, i.e., when also p > 1 is allowed (Theorem 3).

(4,q s +4+1,442+4-2,1,1) PMC C,. This can be continued

Theorem 2: Let q be a prime power and n , p E N.

a) If

n = (p4' - 1) /( 4 - I ) , for some i E NU (0 ) , (6)

then a linear ( 4 , n , q"-i , 1,p) PMC exists. b) If

n=(poq i -1 ) / (q - l ) , for some i E N U { O } and poEN,

with palp and p 4'po,

(7)

then a ( 4 , n , ( p / ~ , , ) q " - ~ , 1, p ) PMC exists.

Proofi a) If p = 1, then i 2 1 (since n 2 1). The sequence {CirSl

defined in the second paragraph of this section does the job then. If p > 1, then we define a similar sequence (Cir=l of PMC's, but we add a CO in front. By (6), p - 1 = p4' - 1 = (4 - 1)n = 0 (mod 4 - 1). Hence there is an integer no such that p = n0(4 - 1)+ 1. Since p > 1, we have no 2 1. Define CO := f f ; O ,

a linear (4,no,4"0,l,p) PMC. Now apply Construction 6) of Section I1 to CO, call the resulting code C,, apply 6) to Cl, call the resulting code C,, etc. (just as in the case p = 1). We get an infinite sequence (Cir=o of linear ( 4 , ni, Mi, 1,p) PMC's over E,, where ni = ( ~ 4 ~ - l)/(q - 1) and Mi = 4"1-'. Indeed, no =

(p4' - l)/(q - l), and if ni = (p4' - l)/(q - l), then

ni+ 1 = qn, + 1

= 4 ( p41- 1) / (q - 1) + 1 = ( pqi+l - 1)/(q - 1).

Also, MO = 4"0-', and if Mi = 4 " 1 - ~ , then

M, = 4 ( ~ - 1 ) n , 4 n , - i = p - i = 4 " r + ~ - ( i + l ) . r + l

b) By part a) of this theorem (with po substituted for p), a linear ( 4 , n = ( p o 4 ' - l ) / ( 4 - 1 ) , M = 4 " - ' , 1 , p o ) PMC C over 5, exists. By (7), p / p o ~ { 1 , 2 ; . . , q " / M I . Apply Construction 5) of Section I1 ("shifting") with p' := p / p o . This yields a PMC C' with the desired parameters. 0

Theorem 3:

a) Let 4 be a prime power and n , p E N. A linear ( 4 , n , . , l , p )

b) Let 4 be a prime and n , p E N. A ( q , n , . , l , p ) PMC exists PMC exists if and only if (6) holds.

if and only if (7) holds.

Proof: a) Suppose a linear (4, n, M,l ,p ) PMC exists. Then M = qi

for some j E {O, 1,; . * , n). By (31, qi(l + ( 4 - 1)n) = p4". Conse- quently, n = (p4' - l)/(q - 11, where i := n - j E (0,1; . ., n}. Hence (6) holds. The "if" part is Theorem 2a).

b) Suppose a (4 ,n ,M,l ,p) PMC exists. By (3), M(1+(4 - 1)n) = p4", hence 1 +(q - l)n = poq', for some po E N and iE(O,l;..,n} with po(p. Also, p o 4 ' = p 4 " / M 2 p . The ''if"

0

Theorem 3 and the proof of Theorem 2 show that the Con- structions 2), 31, 5) , and 6) of Section 11 are the only construc- tions needed to obtain all possible parameters of PMC's with

= 1, provided 4 is a prime, or when we restrict ourselves to linear PMC's. Theorem 3b) leaves open the cases when q is not a prime. For any 4, prime power or not, the cases with M = 4" are trivial: by (3) and Construction 3) of Section 11, the only possible parameters of PMC's with M = 4" are (4 , n, 4", 1, 1 + (4 - l)n), for n = 1,2,3, . . . . These parameters satisfy (7).

part is a consequence of Theorem 2b).

I 'ij -

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682 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 3, MAY 1991

From now on, let 4 be a prime power. In view of Theorem 2b), one could conjecture that the answer to the following question is affirmative.

Open Problem 1: Is it allowed to replace “prime” by “prime power” in Theorem 3b)?

In studying this problem, we can restrict ourselves to PMC’s with M < 4“ , as previously pointed out. To prove the “only if” part of Theorem 3b), we only used the sphere packing equation (3). To this condition, we can add the condition that n 1 (modq), by (9, which was proved in [2] using properties of Lloyd polynomials. These two necessary conditions are still not sufficient to rule out all parameters n and p which do not satisfy (7): The parameters ( 4 = 4, n = 13, M = 223, r = 1 , p = 5) satisfy (3) and n = 1 (mod 4 ) , but they do not satisfy (7). In fact, this is the “smallest” instance (according to the values of 4 , n, and p, in this order) for which we left the question whether a PMC with r = 1 exists unanswered.

We conclude with a condition (Theorem 4c)) on the parame- ters ( 4 , n , . , l ,p) of a PMC, when they do not satisfy (7). As a consequence, we get a lower bound on the parameter p (Corollary 1). This could be a first step in the investigation of the open problem previously mentioned.

Theorem 4: Let p be a prime, m , n , p, M E N , and 4 = p”. Suppose that a ( 4 , n, M , 1, p) PMC exists. Define A E N, k E N U {O} with p t A by

We have the following.

a) If m J k , then (7) holds. b) If A = 1, then mlk and (7) holds. c) If A > 1, write k = sm + t , where s, t E NU (0) and 0 5 t <

m . Then A = p m - ‘ + ( q - l ) a , for some a E N. In particu- lar, A > p + q - l .

Proof: Equation (8) follows immediately from (3). a) Put i := k / m and po := A. Then pk = 4‘ , and by (81,

n = (p04’ - l ) / ( q - 1). Moreover, i E NU{O), p,, E N, poIpL, and p = ApkM/q” I Apk = A4’ = p04‘. Hence (7) holds.

b) See also [6, Theorem (5.1.1)] or 18, Lemma 34, p. 1811. By (81, we have 1 + ( p ” - 1)n = pk. It follows that pm - llpk - 1, hence mlk (see [8, Lemma 9, p. 1021). By a), (7) holds.

c) Put no:=n. For i : = l ; . . , s define n , : = ( n , - l - l ) / q . We have no E N and 1 + ( 4 - l )no = Apsm+‘. Now if for some i E (0 , l ; . ., s - l}, n, is an integer and 1 + ( 4 - l )n , = hp(S-L)m+‘ , then n, = 1 (mod 4 ) , hence n , + , E Z. Furthermore,

Ap(s-(’cl))m+r. This proves that for i = 0,l; . ., s, n, E Z and 1 + ( 4 - l )n , = Ap(s-’)m+r . Moreover, n, > 1 since A > 1 and p + A. In particular, 1 + (q - l )n, = Ap‘. Hence n, = 1 (mod p‘) . Write n, = up‘ + 1, with a E Z. Since n, > 1, a 2 1. We get Ap‘ = 1 + ( q - l ) n , = l + ( q -1Xap‘ + I ) = 4 + ( 4 - l)ap‘, hence A =

1 + ( 4 - 1)n,+1 = 1 + ( 4 - 1Xn, - l ) / q = (1 + ( 4 - l )n , ) /q =

p m - * + ( 4 -1)a. U

Remark 2: In Theorem 4, when A > 1 and m 4 k , it is possible that (7) holds, nevertheless. Example: By Theorem 2a), with i = 1, a (linear) ( 4 = 4, n = 13, M = 412, r = 1, p = 10) PMC exists. Certainly (7) holds; namely, choose i = 1 and po = p = 10. But also (8) is satisfied with A > 1 and m 4 k , if we choose A = p /2 = 5 and k = 3: 1 + (4 - 1)13 = 1O.4l3/4l2 = 5 .Z3.

Corollary I: Let p be a prime, m , n, p E N , and q = p“. If a ( 4 , n, ., 1, p ) PMC exists, and these parameters do not satisfy (7), then p > p + q - l .

Remark 3: We have been informed by one of the (anony- mous) referees that a (4,13, 223, 1,5) PMC exists. Therefore, the answer to open problem 1 is negative. This 5-fold PMC, while obviously not linear over F4, is linear over E,.

ACKNOWLEDGMENT

The authors thank the referees for their helpful comments. S. N. Litsyn would like to express his gratitude to ENST-Paris and the Eindhoven University of Technology for their hospital- ity during his visits in September 1989 and June 1990.

REFERENCES

[ l ] V. M. Blinovskii, “Bounds for codes in the case of list decoding of finite volume,” Probl. Peredach. Inform., vol. 22, pp. 11-25, 1986. Translated in: Problems of Inform. Transm., vol. 22, pp. 7-19, 1986.

[2] R. F. Clayton, “Multiple packings and coverings in algebraic coding theory,” Thesis, Univ. of California, Los Angeles, 1987.

[31 G. D. Forney, Jr., “Exponential error bounds for erasure, list, and decision feedback schemes,” IEEE Trans. Inform. Theory, vol. 14, no. 2, pp. 206-220, Mar. 1968.

[41 P. J. M. van Laarhoven, E. H. L. Aarts, J. H. van Lint, and L. T. Wille, “New upper bounds for the football pool problem for 6. 7, and 8 matches,” J . Combin. Theory, vol. 52, pp. 304-312, 1989.

[SI J. H. van Lint, “A survey of perfect codes,” Rocky Mountain J . Math., vol. 5, pp, 199-224, 1975.

[61 -, Coding Theory. Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1971.

[71 A. C. Lobstein and G . J. M. van Wee, “On normal and subnormal q-ary codes,” IEEE Trans. Inform. Theory, vol. 35, no. 6, pp. 1291-1295, Nov. 1989.

[8] F. J. Macwilliams and N. J. A. Sloane, The Theory of Error-Cor- recting Codes. Amsterdam: North-Holland, 1977.

Runlength Codes From Source Codes

Kenneth J. Kerpez, Member, IEEE

Abstract-Binary runlength codes, also known as (d,k) codes, are used primarily for magnetic and optical recording. A class of ( d , k ) codes is analyzed. These codes are developed by constructing a lossless source code that maps runlengths into unconstrained binary sequences. The source code is constructed for the maxentropic distribution on runlengths. The inverse of the source code, which outputs runlengths guided toward the ideal maxentropic distribution, is the (d, k ) code. Four types of source codes are investigated for this purpose: Huffman, enumerative, variable-length-to-block, and Elias or arithmetic. The rates of the codes are each proven to converge to the capacity with increasing complexity. The codes are not state dependent and are variable rate except for the fixed-rate enumerative code. A combined sourcdd, k ) code is presented that is based on the arithmetic code.

Index Terms-Runlength code, ( d , k ) code, arithmetic code, modula- tion code, maximum entropy, maxentropic.

Manuscript received December 12, 1988, revised May 25, 1990. This work was supported in part by NSF Grant ECS-8352220, IBM, CDC, and AT&T. This work was presented at the International Symposium on Information Theory, Kobe, Japan, June 19-24, 1988.

The author is with Bell Communications Research, 445 South Street, Morristown, NJ 07960-1910.

IEEE Log Number 9141817.

0018-9448/91/0500-0682$01.00 01991 IEEE