# On the weak order ideal associated to linear codes

## TL;DR

This paper investigates a weak order ideal linked to coset leaders in non-binary linear codes, enabling incremental computation and characterization of key codeword sets with properties related to order and support.

## Contribution

It introduces a novel weak order ideal framework for coset leaders, facilitating the computation and description of leader codewords and related sets in linear codes.

## Key findings

- Defines a weak order ideal for coset leaders
- Enables incremental computation of coset leaders
- Characterizes leader codewords and related sets

## Abstract

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