Bounds for Permutation Arrays under Kendall Tau Metric

TL;DR

This paper develops new algorithms and theorems to improve bounds on the size of permutation arrays under the Kendall-tau metric, which are useful for error-correcting codes and recursive bound computation.

Contribution

It introduces improved lower bounds for permutation arrays and defines (n,m,d)-arrays to facilitate recursive bound calculations.

Findings

New algorithms for lower bounds of P(n,d)

Theorems providing improved bounds for permutation arrays

Introduction of (n,m,d)-arrays for recursive bound computation

Abstract

Permutation arrays under the Kendall-$\tau$ metric have been considered for error-correcting codes. Given $n$ and $d\in [1..\binom{n}{2}]$, the task is to find a large permutation array of permutations on $n$ symbols with pairwise Kendall-$\tau$ distance at least $d$. Let $P(n,d)$ denote the maximum size of any permutation array of permutations on $n$ symbols with pairwise Kendall-$\tau$ distance $d$. New algorithms and several theorems are presented, giving improved lower bounds for $P(n,d)$. Also, $(n,m,d)$-arrays are defined, which are permutation arrays on n symbols with Kendall-$\tau$ distance d, with the restriction that symbols {1...(n-m)} appear in increasing order. Let $P(n,m,d)$ denote the maximum size of any $(n,m,d)$-array. For example, (n,m,d)-arrays are useful for recursively computing lower bounds for $P(n,d)$. Lower and upper bounds are given for $P(n.m,d)$.