On the number of alternating paths in bipartite complete graphs

Let $C \subseteq [r]^m$ be a code such that any two words of $C$ have Hamming distance at least $t$. It is not difficult to see that determining a code $C$ with the maximum number of words is equivalent to finding the largest $n$ such that there is an $r$-edge-coloring of $K_{m, n}$ with the property that any pair of vertices in the class of size $n$ has at least $t$ alternating paths (with adjacent edges having different colors) of length $2$. In this paper we consider a more general problem from a slightly different direction. We are interested in finding maximum $t$ such that there is an $r$-edge-coloring of $K_{m,n}$ such that any pair of vertices in class of size $n$ is connected by $t$ internally disjoint and alternating paths of length $2k$. We also study a related problem in which we drop the assumption that paths are internally disjoint. Finally, we introduce a new concept, which we call alternating connectivity. Our proofs make use of random colorings combined with some integer programs.