Lexicodes over Rings

TL;DR

This paper introduces a method for constructing linear lexicodes over finite chain rings using B-ordering and selection criteria, resulting in optimal and good binary codes, including self-dual codes like the octacode.

Contribution

It presents new greedy algorithms for lexicode construction over rings, producing many optimal and near-optimal codes, including self-dual codes.

Findings

Constructed many optimal codes over rings.

Produced good binary codes meeting Gilbert bound.

Obtained optimal self-dual codes like the octacode.

Abstract

In this paper, we consider the construction of linear lexicodes over finite chain rings by using a $B$-ordering over these rings and a selection criterion. % and a greedy Algorithm. As examples we give lexicodes over $\mathbb{Z}_4$ and $\mathbb{F}_2+u\mathbb{F}_2$. %First, greedy algorithms are presented to construct %lexicodes using a multiplicative property. Then, greedy algorithms %are given for the case when the selection criteria is not %multiplicative such as the minimum distance constraint. It is shown that this construction produces many optimal codes over rings and also good binary codes. Some of these codes meet the Gilbert bound. We also obtain optimal self-dual codes, in particular the octacode.