Improved Lower Bounds on the Size of Balls over Permutations with the Infinity Metric

TL;DR

This paper derives new lower bounds on the size of permutation balls under the infinity metric for large radii, narrowing the gap to upper bounds and improving bounds for error correction and covering codes.

Contribution

It introduces improved lower bounds on permutation ball sizes under the infinity metric for large radii, reducing the asymptotic gap to upper bounds.

Findings

Lower bounds reduce the asymptotic gap to upper bounds to 0.029 bits per symbol.

Improved bounds lead to better error-correcting code packing bounds.

Enhanced upper bounds on the size of optimal covering codes.

Abstract

We study the size (or volume) of balls in the metric space of permutations, $S_n$, under the infinity metric. We focus on the regime of balls with radius $r = \rho \cdot (n\!-\!1)$, $\rho \in [0,1]$, i.e., a radius that is a constant fraction of the maximum possible distance. We provide new lower bounds on the size of such balls. These new lower bounds reduce the asymptotic gap to the known upper bounds to at most $0.029$ bits per symbol. Additionally, they imply an improved ball-packing bound for error-correcting codes, and an improved upper bound on the size of optimal covering codes.