The sizes of optimal constant-composition codes of weight three have been determined by Chee, Ge and Ling with four cases in doubt. Group divisible codes played an important role in their constructions. In this paper, we study the problem of constructing optimal quaternary constant-composition codes with Hamming weight four and minimum distances five or six through group divisible codes and Room square approaches. The problem is solved leaving only five lengths undetermined. Previously, the results on the sizes of such quaternary constantcomposition codes were scarce.
Constant‐weight codes (CWCs) have played an important role in coding theory. To construct CWCs, a K‐GDD (where GDD is group divisible design) with the “star” property, denoted by K‐*GDD, was introduced, in which any two intersecting blocks intersect in at most two common groups. In this paper, we consider the existence of 4‐*GDD(gn)s. Previously, the necessary conditions for existence were shown to be sufficient for g=3, and also sufficient for g=6 with prime powers n≡3,5,7(mod8) and n≥19. We continue to investigate the existence of 4‐*GDD(6n)s and show that the necessary condition for the existence of a 4‐*GDD(6n), namely, n≥14, is also sufficient. The known results on the existence of optimal quaternary (n, 5, 4) CWCs are also extended.
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