Covering radius in the Hamming permutation space

Abstract

Let $\mathcal{S}_n$ denote the set of permutations of $\{1,2,\dots,n\}$. The function $f(n,s)$ is defined to be the minimum size of a subset $S\subseteq \mathcal{S}_n$ with the property that for any $\rho\in \mathcal{S}_n$ there exists some $\sigma\in S$ such that the Hamming distance between $\rho$ and $\sigma$ is at most $n-s$. The value of $f(n,2)$ is the subject of a conjecture by K\'ezdy and Snevily, which implies several famous conjectures about latin squares. We prove that the odd $n$ case of the K\'ezdy-Snevily Conjecture implies the whole conjecture. We also show that $f(n,2)>3n/4$ for all $n$, that $s!< f(n,s)< 3s!(n-s)\log n$ for $1\leq s\leq n-2$ and that \[f(n,s)>\left\lfloor \frac{2+\sqrt{2s-2}}{2}\right\rfloor \frac{n}{2}\] if $s\geq 3$.