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

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... ecade. 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 A code G is said to be an (n, IGI, r, /1) multiple 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.