A family of diameter perfect constant-weight codes from Steiner systems

11/30/2022

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by   Minjia Shi, et al.

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If S is a transitive metric space, then |C|·|A| ≤ |S| for any distance-d code C and a set A, "anticode", of diameter less than d. For every Steiner S(t,w,n) system S, we show the existence of a q-ary constant-weight code C of length n, weight w (or n-w), and distance d=2w-t+1 (respectively, d=n-t+1) and an anticode A of diameter d-1 such that the pair (C,A) attains the code-anticode bound and the supports of the codewords of C are the blocks of S (respectively, the complements of the blocks of S). We study the problem of estimating the minimum value of q for which such a code exists, and find that minimum for small values of t. Keywords: diameter perfect codes, anticodes, constant-weight codes, code-anticode bound, Steiner systems.