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

2010; Institute of Electrical and Electronics Engineers; Volume: 56; Issue: 1 Linguagem: Inglês

10.1109/tit.2009.2034814

1557-9654

Yeow Meng Chee, Son Hoang Dau, Alan C. H. Ling, San Ling,

Cooperative Communication and Network Coding

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.