Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes

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An optimal constant-composition or constant-weight code of weight w has linear size if and only if its distance d is at least 2w-1. When d≥ 2w, the determination of the exact size of such a constant-composition or constant-weight code is trivial, but the case of d=2w-1 has been solved previously only for binary and ternary constant-composition and constant-weight codes, and for some sporadic instances. This paper provides a construction for quasicyclic optimal constant-composition and constant-weight codes of weight w and distance 2w-1 based on a new generalization of difference triangle sets. As a result, the sizes of optimal constant-composition codes and optimal constant-weight codes of weight w and distance 2w-1 are determined for all such codes of sufficiently large lengths. This solves an open problem of Etzion. The sizes of optimal constant-composition codes of weight w and distance 2w-1 are also determined for all w≤ 6, except in two cases.