Abstract
The Johnson-type upper bound on the maximum size of a code of length n , distance d=2w-1 , and constant composition {\overline {w}} is \lfloor \dfrac {\vphantom {R^{.}}n}{w_{1}}\rfloor , where w is the total weight and w_{1} is the largest component of {\overline {w}} . Recently, Chee et al. proved that this upper bound can be achieved for all constant-composition codes of sufficiently large lengths. Let N_{ccc}({\overline {w}}) be the smallest such length. The determination of N_{ccc}({\overline {w}}) is trivial for binary codes. This paper provides a lower bound on N_{ccc}({\overline {w}}) , 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}({\overline {w}}) , for all q -ary constant-composition codes, provided that 3w_{1}\geq w with finite possible exceptions.