Let $K_{n,n}$ denote the complete bipartite graph with n vertices in each partite set and $K_{n,n}+I$ denote $K_{n,n}$ with a one-factor added. It is proved in this paper that there exists an m-cycle system of $K_{n,n}+I$ if and only if $n \equiv 1 (\rm{mod} 2)$, $m \equiv 0 (\rm{mod} 2)$, $4 \leq m \leq 2n$, and $n(n+1) \equiv$ 0 (mod m).

We show an interesting pairwise balanced design (PBD)-closure result for the set of lengths of constant-composition codes whose distance and size meet certain conditions. A consequence of this PBD-closure result is that the size of optimal constant-composition codes can be determined for infinite families of parameter sets from just a single example of an optimal code. As an application, the sizes of several infinite families of optimal constant-composition codes are derived. In particular, the problem of determining the size of optimal constant-composition codes having distance four and weight three is solved for all lengths sufficiently large. This problem was previously unresolved for odd lengths, except for lengths seven and eleven.

Let $K_{n,n}$ denote the complete bipartite graph with n vertices in each bipartition set and $K_{n,n}-I$ denote $K_{n,n}$ with a 1‐factor removed. An m‐cycle system of $K_{n,n}-I$ is a collection T of m‐cycles such that each edge of $K_{n,n}-I$ is contained in a unique m‐cycle of T. In this paper, it is proved that the necessary and sufficient conditions for the existence of an m‐cycle system of $K_{n,n}-I$ are $n\equiv 1 \({\rm mod} \ 2)$, $m\equiv 0 \ ({\rm mod} \ 2)$, $4\leq m \leq 2n$, and $n(n-1)\equiv 0\ ({\rm mod}\ m)$.