New bounds of permutation codes under Hamming metric and Kendall's $\tau$-metric

TL;DR

This paper improves bounds for permutation codes under Hamming and Kendall's tau metrics using graph coloring, advancing theoretical limits and narrowing existing gaps for large code lengths.

Contribution

It introduces new bounds for permutation codes under two metrics, notably improving the asymptotic Gilbert-Varshamov bound and refining bounds under Kendall's tau.

Findings

Asymptotic improvement of Gilbert-Varshamov bound by a factor of n under Hamming metric.

Narrowing the gap between known bounds for Kendall's tau-metric.

Some specific results for small parameters under Kendall's tau-metric.

Abstract

Permutation codes are widely studied objects due to their numerous applications in various areas, such as power line communications, block ciphers, and the rank modulation scheme for flash memories. Several kinds of metrics are considered for permutation codes according to their specific applications. This paper concerns some improvements on the bounds of permutation codes under Hamming metric and Kendall's $\tau$-metric respectively, using mainly a graph coloring approach. Specifically, under Hamming metric, we improve the Gilbert-Varshamov bound asymptotically by a factor $n$, when the minimum Hamming distance $d$ is fixed and the code length $n$ goes to infinity. Under Kendall's $\tau$-metric, we narrow the gap between the known lower bounds and upper bounds. Besides, we also obtain some sporadic results under Kendall's $\tau$-metric for small parameters.