The covering radii of the 22-transitive unitary, Suzuki, and Ree groups

We study the covering radii of $2$-transitive permutation groups of Lie rank one, giving bounds and links to finite geometry

Covering radius in the Hamming permutation space

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 $s\geq 3$.Comment: 10 pages, 0 figure

Covering Radius of Permutation Groups with Infinity-Norm

The covering radius of permutation group codes are studied in this paper with $l_{\infty}$-metric. We determine the covering radius of the $(p,q)$-type group, which is a direct product of two cyclic transitive groups. We also deduce the maximum covering radius among all the relabelings of this group under conjugation, that is, permutation groups with the same algebraic structure but with relabelled members. Finally, we give a lower bound of the covering radius of the dihedral group code, which differs from the trivial upper bound by a constant at most one. This improves the result of Karni and Schwartz in 2018, where the gap between their lower and upper bounds tends to infinity as the code length grows.Comment: 13 pages, 0 figure