A New Metric on Symmetric Group and Applications to Block Permutation Codes

TL;DR

This paper introduces a new algebraic-geometric metric related to the block permutation metric, enabling the construction of improved permutation codes with better parameters and surpassing existing bounds.

Contribution

It proposes a novel metric linked to the block permutation metric, facilitating advanced algebraic code constructions with improved parameters and bounds.

Findings

Constructed codes that outperform previous results in block permutation metric

Developed an explicit systematic code construction improving systematic code results

Demonstrated codes surpass the Gilbert-Varshamov bound in the new metric

Abstract

Permutation codes have received a great attention due to various applications. For different applications, one needs permutation codes under different metrics. The generalized Cayley metric was introduced by Chee and Vu [4] and this metric includes several other metrics as special cases. However, the generalized Cayley metric is not easily computable in general. Therefore the block permutation metric was introduced by Yang et al. [22] as the generalized Cayley metric and the block permutation metric have the same magnitude. However, the block permutation metric lacks the symmetry property which restricts more advanced algebraic tools to be involved. In this paper, by introducing a novel metric closely related to the block permutation metric, we build a bridge between some advanced algebraic methods and codes in the block permutation metric. More specifically, based on some techniques from algebraic function fields originated in [19], we give an algebraic-geometric construction of codes in the novel metric with reasonably good parameters. By observing a trivial relation between the novel metric and block permutation metric, we then produce non-systematic codes in block permutation metric that improve all known results given in [21, 22]. More importantly, based on our non-systematic codes, we provide an explicit and systematic construction of codes in block permutation metric which improves the systematic result shown in [22]. In the end, we demonstrate that our codes in the novel metric itself have reasonably good parameters by showing that our construction beats the corresponding Gilbert-Varshamov bound.