  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
$\lfloor\dfrac{n}{w_1}\rfloor$, 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 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.
