On zero neighbours and trial sets of linear codes

TL;DR

This paper investigates the properties of leader codewords in non-binary linear codes, linking them to test sets, trial sets, and zero neighbors through weight-compatible order properties.

Contribution

It introduces a novel characterization of leader codewords and their relation to key sets like test sets and zero neighbors in non-binary linear codes.

Findings

Leader codewords exhibit monotonicity properties in weight-compatible order.

The paper provides descriptions of test sets, trial sets, and zero neighbors using leader codewords.

New theoretical insights into the structure of non-binary linear codes.

Abstract

In this work we study the set of leader codewords of a non-binary linear code. This set has some nice properties related to the monotonicity of the weight compatible order on the generalized support of a vector in $\mathbb F_q^n$. This allows us to describe a test set, a trial set and the set zero neighbours in terms of the leader codewords.