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 $$hd(A) \ge d$$ . 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 $$n\ge k$$ , 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). These results resolve a case left open in a previous paper (Bereg et al. in Des Codes Cryptogr 86(5):1095–1111, 2018), where it was shown that $$M(q-1, q-3) \ge q^{2} - q$$ and $$M(q,q-3) \ge q^{3} - q$$ 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.