A. Lubotzky, R. Phillips, and P. Samak, “Explicit expanders and the Ramanujan conjectures.” in Proc. 18th ACM Symp. Theory of Compur., 1986, 240-246; See also: “Ramanujan graphs,” Combi- narorica, vol. 8, pp. 261-277, 1988. F. J. MacWilliams and N.J.A. Sloane, The Theory of Error-Cor- recling Codes. Amsterdam: North Holland, 1977. G. A. Margulis, “Explicit group-theoretical constructions of combi- natorial schemes and their application to the design of expanders and superconcentrators,” Prob. Inform. Transm., vol. 24, pp. 39-46, 1988. J. Naor and M. Naor, “Small-bias probability spaces: efficient con- structions and applications,” in Proc. 22nd ACM Symp. Theory of Comput., 1990, pp. 213-223. A. Nilli, “On the second eigenvalue of a graph,” Discrete Math.,

J . Riordan, Combinatorial Identities. New York: John Wiley, 1968. P. Sarnak, private communication. G. Seroussi and N . H. Bshouty, “Vector sets for exhaustive testing of logic circuits,” IEEE Trans. Inform. Theory, vol. 34, pp. 513-522, May 1988. Y. Sugiyama, M. Kasahara, S. Hirasawa, and T. Namekawa, “A modification of the constructive asymptotically good codes of Justesen for low rates,” Inform. Contr., vol. 25, pp. 341-350, 1974. - , “A new class of asymptotically good codes beyond the Zyablov bound,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 198-204, Mar. 1978. - , “Superimposed concatenated codes,” IEEE Trans. Inform. Theory, vol. IT-26, pp. 735-736, Nov. 1980. M. A. Tsfasman, S . G. VIHdut, and Th. Zink, “Modular curves, Shimura curves, and Goppa codes, better than Varshamov-Gilbert bound,” Math. Nachr., vol. 109, pp. 21-28, 1982. A. Weil, “Sur les courbes algibriques et les variistis qui sen dkduisent,” Actualit& Sci. Ind., p. 1041, 1948. E. J. Weldon, Jr., “Justesen’s construction-The low-rate case,” IEEE Trans. Inform. Theory, vol. IT-19, pp. 711-713, Sept. 1973. - , “Some results on the problem of constructing asymptotically good error-correcting codes,” IEEE Trans. Inform. Theory, vol.

V. V. Zyablov, “An estimate of the complexity of constructing binary linear cascade codes,” Probl. Inform. Transm., vol. 7, pp. 3-10, 1971.

vol. 91, pp. 207-210, 1991.

IT-21, pp. 412-417, July 1975.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 2, MARCH 1992

On Generator Matrices of Codes

Juriaan Simonis

Abstract-The if class of the q-ary linear codes of given length, dimension and minimum weight is nonempty, it is shown to contain a code whose generator matrix consists of words of minimum weight.

Index Terms-Linear codes, minimum weight, generator matrix.

Recently, Dodunekov [l] showed that any q-ary linear code of length n , dimension k , and minimum weight d possesses a genera- tor matrix consisting of codewords of weight 5 d + t , where

a nonnegative integer in virtue of the Griesmer bound. A perhaps more accessible reference is Theorem 2.6 in Dodunekov and Manev’s paper [2] which, however, is restricted to the binary case.

Manuscript received March 7, 1991. The author is with the Faculty of Technical Mathematics and Informatics,

Delft University of Technology, Mekelweg 4, P.O. Box 5031, 2600 GA Delft, The Netherlands.

IEEE Log Number 9104809.

The following theorem, in a way a strengthening of Dodunekov’s result, may prove useful in nonexistence proofs for linear codes.

Theorem: Any linear code V C F; of dimension k and mini- mum weight d can be transformed into a code 9 C ’$: with the same parameters such that 9 possesses a basis of weight d vectors.

Proof: In the sequel, the linear subspace of F: spanned by a set of vectors x, y ; . . , z E F: will be denoted by (x, y , . . . , t). Let { a , , a2; . . , a, ] C V be a maximal set of independent code- words of weight d. Suppose that t < k . All codewords in the complement of the span ( a , , a , , * . . , a,) of the a, have weight > d, Pick a codeword b , $ ( a , , a , , . . . , a,) of lowest weight, say d, and extend { a , , a , ; .., a,, b , } to a basis { a , , a , ; . ., a, , b , ; . . , b k - / } of the code t?. Now change b , into a vector b; of weight d by changing d - d of the nonzero coordinates into zero ones. Then, linear subspace

is a code of minimum weight d , because ( a , , a 2 ; . ., a , ) is unal- tered aFd the words of V \ ( a , , a 2 , . . . , a,) have changed in at most d - d coordinates. We claim that the dimension of %“ is equal to k . For if dim V‘ < k , then b; would be a linear combina- tion of the vectors a , , a 2 ; . . , a, , b 2 ; . . , b k - < , and, thus, would be an element of the original code V. Since the weight of both b; and b , - b; is smaller than d , these vectors would in fact be contained in the linear subspace ( a , , a2, . . . , a,) which contradicts the factthat b , E V \ ( a , , a , ; . . , a , ) . So %“has the same param- eters as 27 has, but the maximum number of independent weight d codewords in V’ exceeds that of K. The induction process is obvious. 0

Remark: As one of the referees observed, it may happen that the resulting code Y is contained in a coordinate hyperplane of F“ and thus has effective length < n. An example is the code V C IT: with

generator matrix O O [o 1 1 1 -

REFERENCES S. D. Dodunekov, “Zamechanie o vesovoy strukture porozh- dayushchikh matrits lineinykh kodov,” Probl. peredach. inform., vol. 26, pp. 101-104, 1990. S . D. Dodunekov and N. L. Manev. “An improvement of the Griesmer bound for some small minimum distances,” Discrete Appl. Math., vol. 12, pp. 103-114, 1985.

On Binary Cyclic Codes of Odd Lengths from 101 to 127

Dieter Schomaker and Michael Wirtz

Abstract-All binary cyclic codes of odd lengths are checked from 101 to 127 to find codes which are better than those in a table by Verhoeff. There are five such cases, namely, 1117, 36, 321, 1117, 37, 291, 1117, 42, 261, 1117, 49, 241, and 1127, 36, 351 cyclic codes. According to Verhoeffs table the previously known ranges of the highest minimum-

Manuscript received January 3 I , 199 1 ; revised August 8, 1991. The authors are with the Mathematisches lnstitut der Universitat Munster,

IEEE Log Number 9105484. Einsteinstrane 62, D-4400 Munster, Germany.

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 2, MARCH 1992 517

TABLE I NUMBER OF NONTRIVIAL CYCLIC CODES

n no. of codes n no. of codes

101 0 103 2 105 9416 107 0 109 4 111 12 113 8

115 24 117 776 119 148 121 0 123 72 125 0 127 29200

TABLE I1 INTERESTING CYCLIC CODES

so n k bch zeros d wd * 117 37 1361

117 42 117 49 [48] 117 49 [48] 117 48 127 36 1351 127 35 127 36 [35] 127 35

*

*

* *

15 10 18 11 8

24 20 18 22

[O,] 1,3,7,9,13,17,29,39 29 1321 1 0,1,3,5,9,17,29,39 26 [0,11,3,5,7,9,13,39 24 1241 [0,]1,3,7,9,13,17,39 24 [24] 0,1,3,9,13,17,29,39 24

[O,] 1,3,5,7,9,11,13,15,19,29,43,55,63 35 [36] 11 0,1,3,5,7,9,11,13,15,21,23,29,3 1,43 36 IV

[O,] 1,3,5,7,9,11,13,19,21,27,43,47,63 35 1361 111 0,1,3,5,7,9,11,13,19,21,31,47,55,63 36 V

distance were 28-40, 28-40, 25-37, 22-32, and 32-46, respectively. Applying constructions X and Y1 we found [120, 37, 321 and (108, 28, 321 codes. Moreover, the highest minimum-distances that cyclic codes of length 127 can attain are determined.

Index Terms-Binary cyclic codes, minimum-distance, computer search, weight distribution, construction X.

The table of Verhoeff in [7] (previous version published in [8]) gives upper and lower bounds on the highest minimum-distance of binary linear codes up to length 127. This table includes the results of Chen [ 11 and Promhouse and Tavares [5] on the true minimum- distance of all cyclic codes of odd lengths up to 65 resp. of odd lengths from 69 to 99. In [2, p. 1521 Conway and Sloane remark that it would be useful to have a list of the best cyclic codes of length 127. So we decided to investigate the minimum-distance of cyclic codes fitting into the table and not covered by [ l ] and [5]. Our method was as follows.

In the first step, we made a list of all nontrivial cyclic codes (i.e., the generator polynomial g ( X ) does not divide X m - 1 for all m < n) of a specified length n in terms of their zeroes, taking into account degeneracy and equivalence of cyclic codes as described in [4, p. 223f, p. 234fl. Table I gives for each n the number of codes in this list.

In the second step, we constructed for each code a parity-check matrix. In the third step a probabilistic algorithm adapted from Omura [6] (incorporating ideas of linear programming) was used to search for codewords of low weight. This search was stopped as soon as the lowest weight d, found so far was smaller or equal to the lower bound d , from [7]. If, however, d, was still greater than d , after a certain amount of computation, the code was marked for further investigations. The codes surviving this test were of length 117 resp. 127 and of dimension 36, 37, 42, 48, 49 resp. 35, 36. In the fourth step, we computed the weight distribution of the surviv- ing codes of dimension 36 resp. 35 and with the highest d,. These codes C are even-weight subcodes of corresponding codes C‘ of dimension 37 resp. 36. Let A ; denote the number of codewords in C of weight i. The distribution { A : } of C’ can be computed from that of C by the formula A: = A ; + An- , .

TABLE 111 WEIGHT DISTRIBUTION OF THE [117, 37, 291 CODE

32/85 36/81 40177 44/73 48/69 52/65 56/61 60157 64/53 68 149 72/45 76/41 80137 84/33 88/29

92079 3050580

55402074 552871332

3091147371 9874063368

18196458228 19478223336 12106352232 4354278084

896552202 104057460

6702813 220896

4680

We were unable to compute the weight distribution of the codes of dimension 42, 48, and 49 (for dimension 42 the computing time on a MC68020-based SUN 3 we used would be more than 900 hours). In these cases, in a fifth step, the minimum-distance was found by the method suggested by Coppersmith and Seroussi [3].

Table I1 shows for each length and dimension the surviving codes with the highest d, (in all cases dr was equal to the true minimum-distance d). For reasons of space only some codes of length 127 are listed. If a code occurred together with its even-weight subcode, they were combined in one entry.

In Table I1 self-orthogonal codes are marked with a star in column “so” (since n is odd this is only a property of the lower-dimensional code). The entry in column “wd” refers to the weight distribution of the lower dimensional code in the same row (see Appendix) and column “bch” contains the BCH-bound for this code. For example, the first entry in Table I1 means that the weight distribution denoted by I belongs to the [117, 361 code C with zeroes 0, 1, 3, 7, 9, 13, 17, 29, 39. The weight distribution of the corresponding [117, 371 code C‘ with zeroes 1, 3, 7, 9, 13, 17, 29, 39 is shown in Table 111.

Thus, we have found codes with the following parameters im-

518 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 2, MARCH 1992

proving on Verhoeffs table: [117, 36, 321, [117, 37, 291, [117, 42, 261, [117, 49, 241, [127, 36, 351. Using the inclusion between the first two of these codes, Construction X (see [4, ch. 181) yields a new [120, 37, 321 code. Since the dual of the 1117, 36, 321 code has minimum-distance 9 it can be shortened to a [log, 28, 2 321 code (Construction Y I ) . As a result 140 lower bounds in [7] are raised and all gaps between lower and upper bound of size 14, 15, and 16 (see [7, Table VI) are eliminated. Inspection of Verhoeffs table and Fig. 9.1 in [4] shows that for all possible dimensions k of cyclic codes of length 127 (i.e., k = 0 or 1 (mod 7)) the lower bounds come from (even-weight subcodes of) narrow-sense BCH-codes, with one exception at k = 35/36. There are several, inequivalent cyclic codes to fill this gap, see Table 11. Moreover our search has established the following fact.

Fact: There are no binary cyclic codes of length 127 and dimen- sion k # 35, 36 with a higher minimum-distance than the (even- weight subcodes of) narrow-sense BCH-codes of the same dimen- sion. For binary cyclic [127, 351 (resp. [127, 361) codes the highest minimum-distance is 36 (resp. 35).

ACKNOWLEDGMENT

The authors would like to thank Prof. W. Scharlau for suggesting the subject of this correspondence.

APPENDIX

I: i Ai 11: i

32 36 40 44 48 52 56 60 64 68 72 76 80 84 88

92079 3050580 55402074 552871332 3091147371 9874063368 18196458228 19478223336 12106352232 4354278084 896552202 104057460 67028 13 220896 4680

LII: i

36 40 44 48 52 56 60 64 68 72 76 80 84 88 92

Ai /I27 33 1

11452 18265 1 1722742 95840 15 31677928 63152943 76164793 557 18481 24644732 6552833 1035538 95381 5152 149

36 40 44 48 52 56 60 64 68 72 76 80 84 88 92

IV: i

36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66

-

A, 1127 535

11 172 181671 1724954 9580879 31687436 63135751 76179297 55709661 24654252 6545021 1037190 96317 4900 85

A,/127

V: i A, 1127 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96

400 1 1249 182056 1726459 9576756 31685789 63 1441 I6 76179670 55699224 24658067 6550096 1033193 97124 4767 148

7

68 70 72 74 76 78 80 82 84 86 88 90 92 94

28071414 19588356 12420436 6717914 3303104 1381212 520828 167398 48304 11634 2562 448 86 14

REFERENCES

[l] C. L. Chen, “Computer results on the minimum distance of some binary cyclic codes,” IEEE Trans. Inform. Theory, vol. IT-16, pp. 359-360, May 1970. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups. New York: Springer, 1988. D. Coppersmith and G. Seroussi, “On the minimum distance of some quadratic residue codes,” IEEE Trans. Inform. Theory, vol. IT-30, pp. 407-411, Mar. 1984. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error- Correcting Codes. Amsterdam: North-Holland, 1977. G. Promhouse and S. Tavares, “The minimum distance of all binary cyclic codes of odd lengths from 69 to 99,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 438-442, July 1978.

[6] J. K. Omura, “Iterative decoding of linear codes by a modulo-2 linear program,” Discrete Math., vol. 3, pp. 193-208, 1972.

(71 T. Verhoeff, “An updated table of minimum-distance bounds for binary linear codes,” preprint, Jan. 1989.

[8] -, “An updated table of minimum-distance bounds for binary linear codes,” IEEE Trans. Inform. Theory, vol. IT-33, pp. 665-680, Sept. 1987.

[2]

[3]

[4]

[SI

Disjoint Difference Sets, Difference Triangle Sets, and Related Codes

Chen Zhi, Fan Pingzhi, and Jin Fan 238 1120 5488 24402 92036 297360 869624 2 15 I678 4826892 9216522 15964186 23630068 3 182 I5 10 36626982 38384685 34402620

Abstract-Disjoint difference sets (DDS), difference triangle sets (DTS), and related codes are discussed and a recursive construction for DDS is given. With this construction and the relationship between DDS and DTS, many new upper bounds for DTS and some better orthogonal codes are obtained.

Index Terms-Combinatorial mathematics, quasi-cyclic codes, convo- lutional codes, optical orthogonal codes.

I. INTRODUCTION

Disjoint difference sets (DDS) were used to construct self-or- thogonal, quasi-cyclic codes by Townsend and Weldon [I]. Differ- ence triangle sets (DTS) were used to construct convolutional

Manuscript received January 24, 1991. The authors are with the Department of Computer Science and Engineer-

IEEE Log Number 9104810. ing, Southwest Jiaotong University, Chengdu, Sichuan 610031, P.R.C.

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