New Lower Bounds for Permutation Arrays Using Contraction

TL;DR

This paper introduces new lower bounds for the maximum size of permutation arrays with given Hamming distance, using contraction operations on sharply transitive groups, advancing understanding of permutation array bounds.

Contribution

It presents novel lower bounds for permutation array sizes by applying contraction to sharply transitive groups, resolving previously open cases.

Findings

New bounds for prime power q with q ≡ 1 (mod 3)

Bounds for permutation arrays from Mathieu groups

Improved lower bounds for specific (n, d) pairs

Abstract

A permutation array $A$ is a set of permutations on a finite set $\Omega$, say of size $n$. Given distinct permutations $\pi, \sigma\in \Omega$, we let $hd(\pi, \sigma) = |\{ x\in \Omega: \pi(x) \ne \sigma(x) \}|$, called the Hamming distance between $\pi$ and $\sigma$. Now let $hd(A) =$ min$\{ hd(\pi, \sigma): \pi, \sigma \in A \}$. For positive integers $n$ and $d$ with $d\le n$, we let $M(n,d)$ be the maximum number of permutations in any array $A$ satisfying $hd(A) \geq d$. There is an extensive literature on the function $M(n,d)$, motivated in part by suggested applications to error correcting codes for message transmission over power lines. A basic fact is that if a permutation group $G$ is sharply $k$-transitive on a set of size $n\geq k$, then $M(n,n-k+1) = |G|$. Motivated by this we consider the permutation groups $AGL(1,q)$ and $PGL(2,q)$ acting sharply $2$-transitively on $GF(q)$ and sharply $3$-transitively on $GF(q)\cup \{\infty\}$ respectively. Applying a contraction operation to these groups, we obtain the following new lower bounds for prime powers $q$ satisfying $q\equiv 1$ (mod $3$). 1. $M(q-1,q-3)\geq (q^{2} - 1)/2$ for $q$ odd, $q\geq 7$, 2. $M(q-1,q-3)\geq (q-1)(q+2)/3$ for $q$ even, $q\geq 8$, 3. $M(q,q-3)\geq Kq^{2}\log q$ for some constant $K$ if $q$ is odd, $q\geq 13$. These results resolve a case left open in a previous paper \cite{BLS}, where it was shown that $M(q-1, q-3) \geq q^{2} - q$ and $M(q,q-3) \geq q^{3} - q$ for all prime powers $q$ such that $q\not \equiv 1$ (mod $3$). We also obtain lower bounds for $M(n,d)$ for a finite number of exceptional pairs $n,d$, by applying this contraction operation to the sharply $4$ and $5$-transitive Mathieu groups.