On the ideal associated to a linear code

TL;DR

This paper establishes a connection between the algebraic structure of linear codes and decoding by associating a binomial ideal, enabling the derivation of test-sets and decoding procedures through Gröbner bases.

Contribution

It introduces a specific binomial ideal linked to a linear code and shows how Gröbner and Graver bases of this ideal facilitate decoding and identify minimal support codewords.

Findings

Reduced Gröbner basis induces a test-set for decoding.

Graver basis provides a universal test-set including minimal support codewords.

The approach bridges algebraic structures with decoding algorithms.

Abstract

This article aims to explore the bridge between the algebraic structure of a linear code and the complete decoding process. To this end, we associate a specific binomial ideal $I_+(\mathcal C)$ to an arbitrary linear code. The binomials involved in the reduced Gr\"obner basis of such an ideal relative to a degree-compatible ordering induce a uniquely defined test-set for the code, and this allows the description of a Hamming metric decoding procedure. Moreover, the binomials involved in the Graver basis of $I_+(\mathcal C)$ provide a universal test-set which turns out to be a set containing the set of codewords of minimal support of the code.