370 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 2, MARCH 1991

(yo , . . . ,y , , - l ) , where yi=tfci)?-1-e(92Pi-l)e+1, O i i < n - l , and y,, - = lf(“ - I)( 17 - - lie+ (f(i) is defined as in Theorem 2). Then 8 = CG,(n,a, y ) .

Proof: a and y are the same as in Theorem 3 (put k = e + 2). Hence, a and y define a CG,(n ,a ,y ) code over GF(q). To show the identity 8 = CG,(n,a,Y), it suffices to prove that the zeroes of the two codes coincide. As in the proof of Theorem 3, we notice that yiai, 0 I i -< n - 1 and r E (0, 1, e, e + 1) give us all the entries of the generator matrix (13) of CG,(n ,a ,y ) , including those of the singular column. The rows of the genera- tor matrix (13) of CG,(n, a , y ) are given by

c r ( x ) = yp jx ’ , r E ( O , l , e , e + I ) n - 1

i = O

where “ - 1

similar to (4) and (5). Because of the restrictions on r , we only have to find the zeroes of (17) when

s + t E(O, l ,e ,e+l} . (18) Otherwise, one of the binomial coefficients of (16) is zero.

Put x [ r d q + 1 ) / n + J ( q 2 - 1 1 ) / n in (17). Then,

1’. S ( C I o ( s + l ) / n + 1 ( 4 2 - 1)/n ) = ncl ( t ( q Z - 1 ) / 2 n ( e + l - r , - Z ( s + 1 ) + 2 J )

r = O

(19)

By using Fact 5, it is easily deduced that (19) is zero if

n - e - 1 j$- 2 + j , + ( s + t ) m o d n . (20)

Combining (18) and (20) give us that the zeroes of (161, and thereby the zeroes of CG,(n, a, y ) , are identical to the zeroes of 8 (15). 0

Corollary: Any [ q + 1,4] and [ q + 1, q - 31 MDS code over GF(q), q = 2h, h > 3, is equivalent to a cyclic code.

Proof: According to Theorem 4, any [ q + 1,4] MDS code 8 over GF(q), q = 2h, h > 3, is equivalent to CG,(q + 1,a, y) code. Since all the elements of GF(4) are represented in a, 8 is in particular equivalent to a CGe(q + 1, a, y ) code, where a and y are defined as in Theorem 5 with n = q + 1 and io = 0 in f(i), i.e., a = 1. Now, the proof of Theorem 5 yields that the zeroes of this code are identical to the zeroes of a cyclic code with zero-set as in (15) (again n = q + 1 and a = 1). The case of [q + 1, q - 31 MDS codes follows from duality (Facts 3 and 4).

VI. CONCLUSION In this correspondence we have studied the pseudo-cyclic

MDS codes constructed by Krishna and Sarwate. These codes, which are extensions the BCH-codes, turn out to be GRS codes. Especially interesting are the codes of length q + 1, correspond- ing to the MDS-conjecture. However, these pseudo-cyclic codes are not the only ones that are MDS. The [q + 1,41 and [q + 1, q -31 codes, q = 2h, are examples of non-GRS pseudo-cyclic MDS codes. Thus, it is still an open problem to determine

whether the pseudo-cyclic [n, k ] MDS codes constructed by Krishna and Sarwate are the only pseudo-cyclic MDS codes with these parameters.

REFERENCES

[l] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Cor- recting Codes. Amsterdam, The Netherlands: North-Holland, 1981.

[2] A. Diir, “The automorphism groups of Reed-Solomon codes,” J . Comb. Theory, Series A, vol. 44, pp. 69-82, 1987.

[3] G. Seroussi and R. M. Roth, “On MDS extensions of generalized Reed-Solomon codes,” IEEE Trans. Inform. Theory, vol. IT-32, no. 3, pp. 349-354, May 1986.

[4] R. M. Roth and G. Seroussi, “On generator matrices of MDS codes,” IEEE Trans. Inform. Theory, vol. IT-31, no. 6, pp. 826-830, Nov. 1985.

[5] R. M. Roth and A. Lempel, “On MDS codes via Cauchy matrices,” IEEE Trans. Inform. Theory, vol. 35, no. 6, pp. 1314-1319, NOV. 1989.

[6] E. R. Berlekamp, Algebraic Coding Theory. New York: McGraw- Hill, 1968.

[7] A. Krishna and D. V. Sarwate, “Pseudocyclic maximum-distance- separable codes,” IEEE Trans. Inform. Theory, vol. 36, no. 4, pp.

[8] V. C. da Rocha, “Maximum distance separable multilevel codes,” IEEE Trans. Inform. Theory, vol. IT-30, no. 3, pp. 547-548, May 1984.

[9] E. Zehendner, “A non-existence theorem for cyclic MDS-codes,” Atti Sem. Mat. Fk. Uniu. Modena, vol. XXXII, pp. 203-205, 1983.

[lo] R. M. Roth and G. Seroussi, “On cyclic MDS codes of length q over GF(q),” IEEE Trans. Inform. Theory, vol. IT-32, no. 2, pp. 284-285, Mar. 1986.

[ l l ] H. Hirschfeld, “Ovals in Desarguesian planes of even order,” Ann. Mat. Pura Appl., vol. 102, pp. 79-89, 1975.

[12] L. R. A. Casse and D. G. Glynn, “The solution to Beniamino Segre’s problem I r ,q , r = 3, q = 2h,” Geom. Ded., vol. 13, pp.

[13] D. G. Glynn, “The non-classical 10-arc of PG(4,9),” Discrete

[141 A. Diir, “On linear MDS codes of length q + 1 over GF(q) for even

[151 C. Dah1 and J. P. Pedersen, “Cyclic and pseudo-cyclic MDS codes

880-884, July 1990.

157-163, 1982.

Math., vol. 59, pp. 43-51, 1986.

4,” J. Comb. Theory, Series A, vol. 49, pp. 172-174, 1988.

of length q + 1,” J . Comb. Theory, Series A, to appear.

Some New Constant Weight Codes

Klaus-Uwe Koschnick

Abstract -An algorithm for the construction of constant weight codes

Index T e m -Constant weight codes, combinatorial optimization.

and some results of applications of this algorithm are presented.

I. INTRODUCTION We denote by A ( n , d , w ) the maximum cardinality of any

binary code of length n, minimum distance d , and constant weight w. In this correspondence we state the results of our search for good constant weight codes using a combinatorial optimization algorithm.

We start with a brief description of our algorithm and then present the results.

Manuscript received August 25, 1989; revised August 28, 1990. The author is with the Universitat Bielefeld, Fakultat fur Mathematik,

IEEE Log Number 9040743. Postfach 8640, 4800 Bielefeld 1, West Germany.

0018-9448/91/0300~0370$01.00 01991 IEEE

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 2, MARCH 1991 371

11. THE ALGORITHM One way of obtaining lower bounds for A(n, d, w) is to formu-

late the problem of constructing constant weight codes as a combinatorial optimization problem and solving this problem with an appropriate algorithm. Examples for this approach can be found in [2] and [3].

We have mapped the problem of constructing a constant weight code of length n, minimum distance d , and weight w containing M codewords to the optimization problem defined as follows: Let R, the set of configurations, be the set of all sequences (x; + ., x,) of M binary vectors of length n and weight w, and the cost c ( X ) of a configuration X = (x1; . ., x,) be defined by

C( X ) = ( d - d ” ( X i 7 . j ) ) 9

l s i < j s M d H ( X , , X j ) < d

where d , denotes the Hamming distance. Minimize c ( X ) on R. In order to solve such an optimization

problem for given parameters n, d , w, and M , we have used the following algorithm.

{Initialization}

1) Randomly choose a configuration from R as current

2) Set count := 0 configuration X.

and backstep-count := 0.

@orward Search}

3 ) IF C(x)= 0 THEN STOP. 4) IF count > limit-1 THEN GOTO Step 12). 5) Build a list L of all words of X that are at a distance

6) Randomly select a codeword x from L. 7 ) Delete x from the list L. 8) Set

d’ < d from some other codeword.

Cmin .- .- min c ( X ’ ) , X‘E N ( X , X )

where N ( X , x ) denotes the set of all configurations that can be obtained by a transposition of a 0 and a 1 in x.

9) IF C,in < C ( X ) THEN set count := 0 ELSE increment count by 1.

THEN randomly choose one of the configurations X E N( X , x ) with c( X ’ ) = cmin as new current configuration X and GOTO Step 3).

10) IF Cmin S C(x)

11) IF the list L is not empty THEN GOTO Step 6).

Backstep}

12) Increment backstep-count by 1. 13) IF backstep-count > limit-2

14) Set delta:=Z, where Z is random variable that takes THEN STOP.

values in the set of even natural numbers with

prob ( Z = 2k) = (1 - P ) ~ - ‘ p , for k = 1,2, . . ,

for some constant p (we have chosen p = 0.75). 15) Iteratively inspect configurations obtained by small ran-

dom perturbations of the current configuration until a configuration with cost c ( X ) + delta is found. Then ac-

TABLE I PARAMETERS OF SOME NEW CONSTANT

WEIGHT CODES

n d W M 12 4 5 80 17 8 7 22 18 8 6 21 22 10 7 16 22 10 8 23 22 10 9 32 23 10 7 20 23 10 9 40 24 10 8 34

TABLE I1 SIZE 20 CONSTANT WEIGHT CODE FOR A(23,10,7)

1) 001OOO101oO0110Ooo1010 11) ooo11o0o100000101000011 2) OO1111000101OOOOOOOO100 12) 011010011OOOOOOOO101Ooo

5 ) 00110000001001011001OOO 15) 0101OOO1OOOOOOO1OO10110

3) 01OOO100111OOOOO101OOOO 13) 1100101OO1OOO10100 4) OOO00111M)0001101OOO100 14) lOlOOOOlOllOOOlO10

6) OOOOOOOO1lOOlOllOOOll00 16) 11100000000010001000101 7) 1OOOOOOO101101OOO1OO1OO 17) oooO1010001OOOOOOO11101 8) OO01OOO1010011OOO101 18) oooO11000010100101010 9) 00100000000100110110001 19) 10101OOO11OOOOO11000

10) OOO000100101o0O01101010 20) 0 1 0 1 0 0 1 o O 0 1 1 1 0 1 0 0

cept this configuration as new current configuration X . If after some time no such configuration has been found, accept the configuration with lowest cost among those of the inspected ones that have cost greater than c ( X ) + delta.

16) Set count := 0. 17) GOTO Step 3).

During the work with this algorithm it turned out that the values in the range from 2.M to 10.M are in most cases appropriate choices for the constant limit-1. limit-2 has ,usually been chosen not greater than 1000.

111. THE RESULTS

The application of the algorithm has led to some new con- stant weight codes. Table I gives their parameters.

After our work had been completed we became aware of newer results of Brouwer, Shearer, Sloane, and Smith [l], whose paper presents a table of constant weight codes of length n I 28 and gives explicit constructions for several hundred new codes. The only case in which our lower bound supersedes the lower bound of [l] for the same parameters, is the case A(23,10,7) 2 20. Our code with these parameters is given in Table 11.

REFERENCES [l] 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,

[2] G. Dueck and T. Scheuer, “Threshold accepting: A general purpose optimization algorithm appearing superior to simulated annealing,’’ Tech. Report 88.10.011, IBM Scientific Center Heidelberg, Oct. 1988.

[3] A. E. Gamal, L. A. Hemachandra, I. Shperling, and V. Wei, “Using simulated annealing to design good codes,” IEEE Trans. Inform. Theory, vol. IT-33, no. 1, pp. 116-123, Jan. 1987.

vol. 36, no. 6, pp. 1334-1380, NOV. 1990.