New Upper Bounds on Sizes of Permutation Arrays

TL;DR

This paper establishes new asymptotic upper bounds on the maximum size of permutation arrays with given length and distance, improving previous bounds for certain parameter regimes.

Contribution

It introduces novel upper bounds on permutation array sizes for specific growth conditions of the distance parameter, advancing theoretical understanding.

Findings

New asymptotic upper bounds for P(n,d)

Bounds are better than previous ones for certain d=n^β

Applicable for constant α, β with specific conditions

Abstract

A permutation array(or code) of length $n$ and distance $d$, denoted by $(n,d)$ PA, is a set of permutations $C$ from some fixed set of $n$ elements such that the Hamming distance between distinct members $\mathbf{x},\mathbf{y}\in C$ is at least $d$. Let $P(n,d)$ denote the maximum size of an $(n,d)$ PA. New upper bounds on $P(n,d)$ are given. For constant $\alpha,\beta$ satisfying certain conditions, whenever $d=\beta n^{\alpha}$, the new upper bounds are asymptotically better than the previous ones.