New Lower Bounds for Permutation Arrays Using Contraction

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.