Constructing Permutation Arrays from Groups

TL;DR

This paper improves lower bounds for the maximum size of permutation arrays with given Hamming distance using algebraic, combinatorial, and computational methods, including group analysis and randomized algorithms.

Contribution

It introduces new bounds for M(n,d) by analyzing affine and projective semilinear groups, and develops randomized algorithms and theorems for contraction operations.

Findings

Improved lower bounds for M(n,d) using group-theoretic methods.

Computed Hamming distances for affine and projective semilinear groups.

Established a quadratic lower bound for M(n,n-2) under specific conditions.

Abstract

Let M(n, d) be the maximum size of a permutation array on n symbols with pairwise Hamming distance at least d. We use various combinatorial, algebraic, and computational methods to improve lower bounds for M(n, d). We compute the Hamming distances of affine semilinear groups and projective semilinear groups, and unions of cosets of AGL(1,q) and PGL(2,q) with Frobenius maps to obtain new, improved lower bounds for M(n,d). We give new randomized algorithms. We give better lower bounds for M(n,d) also using new theorems concerning the contraction operation. For example, we prove a quadratic lower bound for M(n,n-2) for all n=2 (mod 3) such that n+1 is a prime power.