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.
Categories: math.CO cs.IT math.IT