SOME NEW RESULTS ON BINARY ASYMMETRIC ERROR-CORRECTING CODES

G . Fang and H. C. A. van Tilborg

Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. box 513, 5600 MB Eindhoven,

T h e Netherlands

Summary

Let C(n, dA,) denote a hinary, asymmetric error-correcting code of length n and asymmetric minimum distance dA,. Such a code is called optimal, if i t has maximum size. This maximum size is denoted by AA,(R, dA.1. It is shown that there exist exactly four non-isomorphic optimal C(3 ,2 ) codes, four non-isomorphic optimal C(5,2) codes and twelve non-iso- morphic optimal C(7,2) codes. Further i t is proved that optimal C(n, 2) codes of length n=2,4,6 and 8 must be unique. The sizes of all theses codes are given in the following table.

n 1 2 1 3 1 4 1 5 1 6 1 7 1 8 Aa,(n,2) 1 2 1 2 1 4 6 I 12 118 I 3 6

Next a number of new constructions of asymmetric error-correcting codes are presented and better upperbounds on their cardinality are derived. For some of these results i t was necessary t o analyse the structure of such a code by means of its weight enumerator. The next table presents these lower and upper hounds on A A , ( ~ , dA,) for 12 5 n 5 27 and 5 5 dA. 5 9.

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

6 8 12 16 26

40-44 54-74

69- 128 104-228 163-423 243-745

4 4 6 7 8 12 16

22-23 32-34 48-60 66- 110 91-210 124-380 173-721

249-1350

2 4 4 4 4 6 7 9 12 14

19-20 27-30 40-46 58-80

80-144

2 2 2 2 2 2 4 2 4 2 4 4 4 4 6 4 6 4 8 4

8-9 6 12 7

13-14 8 18-19 9 23-26 12

Theorem C(n,dA.) codes ( n >_ dA. 2 2) for which all the Goldbaum inequalities ( [ 5 ] )

for all i = 0 , 1 , . . . , n, are sharp must have a trivial form, namely the repetition code or the whole vector space (GF(2))“.

References

[I] J . H. Weber, C. de Vroedt, and D. E. Boekee, “Bounds and con- structions for binary codes of length less than 24 and asymmetric distance less than 6,” IEEE Trans. Inform. Theory, vol. IT-34, pp. 1321-1331, Sept. 1988.

[2] Yuichi Saitoh, Kazuhiko Yamaguchi and Hideki Imai, “Some new binary codes correcting asymmetric/unidirectional errors,” IEEE Trans. Inform. Theory, vol. 36, No. 3, pp. 645-647, May 1990.

[3] M. R. Best, “A(11,4,4)=35or some new optimal constant weight codes,” ZN 71/77 February, C W I report, Amsterdam.

[4] Tuvi Etzion, “New lower bounds for asymmetric and unidirec- tional codes,” preprint.

[5] I. Ya. Goldhaum, “Estimate for the number of signals in codes correcting nonsymmetric errors,” (in Russian) Automat. Tele- mekh., vol. 32, pp. 94-97, 1971 (English translation: Automat. Rem. Control, vol. 32, pp. 1783-1785, 1971).

[6] 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. IT-36, pp. 1334-1380, November 1990.

Further the following miscellaneous results have been obtained, namely

A ~ , ( 1 0 , 2 ) 5 116

and

143