Computing coset leaders and leader codewords of binary codes

TL;DR

This paper introduces efficient algorithms using Gr"obner representations to compute coset leaders and leader codewords of binary codes, aiding decoding and analysis of code properties.

Contribution

It presents novel algorithms for computing coset leaders and leader codewords, and relates zero neighbors to leader codewords, enhancing decoding techniques.

Findings

Algorithms for computing coset leaders and leader codewords

Methods to determine code covering radius and weight distribution

A new relation between zero neighbors and leader codewords

Abstract

In this paper we use the Gr\"obner representation of a binary linear code $\mathcal C$ to give efficient algorithms for computing the whole set of coset leaders, denoted by $\mathrm{CL}(\mathcal C)$ and the set of leader codewords, denoted by $\mathrm L(\mathcal C)$. The first algorithm could be adapted to provide not only the Newton and the covering radius of $\mathcal C$ but also to determine the coset leader weight distribution. Moreover, providing the set of leader codewords we have a test-set for decoding by a gradient-like decoding algorithm. Another contribution of this article is the relation stablished between zero neighbours and leader codewords.