New multiple covering codes by tabu search

The problem of finding multiple coverings words) of the space is considered. of upper and lower bounds for such codes were et aI., Bounds for multiple covering codes, Des. Codes Cryptogr. 3 (1993), The new codes found in this work improve on 27 upper bounds in those tables. The codes were found using tabu search. The of this method is and it is shown how it also can be used to search for large codes. The problem of finding good of Hamming spaces has attracted a lot of attention during the last decade. In this paper binary codes will be discussed. However, many of the results can be to codes over other alphabets. We consider codes over , where = {O, I} is the two-element Galois field. A code is a nonempty set G ~ F:f. In some particular cases we allow G to be a multiset. The rtamrmng distance d(x, y) between two words x, y E Ff is the number of coordinates in which they differ. The distance between a word x and a code G ~ Ff is d(x, G) mincEC d(x, c). A code G is said to be an (n, IGI, r, /1) multiple (Me) if for all x E Ff there is a set of codewords C' ~ G, such that IC'I /1 and d(x, c) r for all c E Cf. Furthermore, if we allow C and G f to be multisets we call the code a m1J,ltiple covering with repeated codewords (MeR). We are now interested in the functions K(n, r, /1) K(n, r, /1) min{M I there is an (n,M,r,/1) MC} and min{M I there is an (n, r, /1) MCR}. It is in practice impossible to determine exact values of these functions in the general case, so effort has been put into obtaining upper and lower bounds. Upper bounds are constructive: they are proved by finding a corresponding code. If /1 1 '"This research was supported by the Academy of Finland. Austt~alasian Journal of Combinatorics 1995); pp.145-155 we are traditional covering codes, which have been studied; the most recent tables of upper and lower bounds on binary covering codes can be found in [15] and [12], respectively. Earlier results on multiple coverings include those by Clayton [3] and Van Wee et ai. [161. Hamalainen et at. [6] collected bounds on K(n, T, JL) and K(n, T, M) for n 16, M 4. At the end of their introduction mention some reasons why the codes in the paper can be considered reasonably good. Anyhow, in this paper it is shown how efficient and extensive computer searches have led to many as 27 on their upper bounds. New results on lower bounds for multiple covering codes can be found in [2]. In Section the optimization method used in the search, tabu search, is briefly discussed. A matrix method that can be used to find codes is and data structures employed to make the search more efficient are explained. It i8 also mentioned how these can be slightly modified and used in search for other types of In Section 3 the new upper bounds tabulated and compared with the best known old results. Codes corresponding to new bounds are listed in the Appendix.