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 Kézdy and Snevily, which implies several famous conjectures about latin squares. We prove that the odd $n$ case of the Kézdy-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 $1\le s\le n-2$ and that \[f(n,s)>\left\lfloor \frac{2+\sqrt{2s-2}}{2}\right\rfloor \frac{n}{2}\] if $s\ge 3$.

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