Linear Size Constant-Composition Codes Meeting the Johnson Bound

The Johnson-type upper bound on the maximum size of a code of length n, distance d = 2w - 1, and constant n composition w̅ is [n/w
₁
] the largest component of w̅. Recently, Chee et al. proved that this upper bound can be achieved for all constant-composition codes of sufficiently large lengths. Let N
_ccc
(w) be the smallest such length. The determination of N
_ccc
(w̅) is trivial for binary codes. This paper provides a lower bound on N
_ccc
(w̅), which is shown to be tight for all ternary and quaternary codes by giving new combinatorial constructions. Consequently, by the refining method, we determine the values of N
_ccc
(w̅), for all q-ary constant-composition codes, provided that 3w
₁
≥ w with finite possible exceptions.