Constructing Permutation Arrays using Partition and Extension

TL;DR

This paper introduces a versatile partition and extension method to construct large sets of permutations with high Hamming distance, improving lower bounds for permutation arrays relevant to error correction in communications.

Contribution

The paper presents a novel, universal partition and extension technique with three variants, along with algorithms to compute partitions, significantly advancing the construction of permutation arrays.

Findings

New lower bounds for M(n,d) up to n=600

Effective algorithms for partition computation

Improved bounds for M(n,n-1) using MOLS

Abstract

We give new lower bounds for $M(n,d)$, for various positive integers $n$ and $d$ with $n>d$, where $M(n,d)$ is the largest number of permutations on $n$ symbols with pairwise Hamming distance at least $d$. Large sets of permutations on $n$ symbols with pairwise Hamming distance $d$ is a necessary component of constructing error correcting permutation codes, which have been proposed for power-line communications. Our technique, {\em partition and extension}, is universally applicable to constructing such sets for all $n$ and all $d$, $d<n$. We describe three new techniques, {\em sequential partition and extension}, {\em parallel partition and extension}, and a {\em modified Kronecker product operation}, which extend the applicability of partition and extension in different ways. We describe how partition and extension gives improved lower bounds for M(n,n-1) using mutually orthogonal Latin squares (MOLS). We present efficient algorithms for computing new partitions: an iterative greedy algorithm and an algorithm based on integer linear programming. These algorithms yield partitions of positions (or symbols) used as input to our partition and extension techniques. We report many new lower bounds for for $M(n,d)$ found using these techniques for $n$ up to $600$.