On Binomial Ideals associated to Linear Codes

TL;DR

This paper explores the algebraic structure of binomial ideals linked to linear codes, connecting different approaches and proposing a new decoding heuristic based on Gröbner bases for codes over prime fields.

Contribution

It establishes connections between two generalizations of binomial ideals associated with linear codes and introduces a novel decoding heuristic using Gröbner bases.

Findings

Code ideals are related by elimination techniques.

A new heuristic decoding method for prime field codes is proposed.

Connections between different algebraic approaches to code ideals are clarified.

Abstract

Recently, it was shown that a binary linear code can be associated to a binomial ideal given as the sum of a toric ideal and a non-prime ideal. Since then two different generalizations have been provided which coincide for the binary case. In this paper, we establish some connections between the two approaches. In particular, we show that the corresponding code ideals are related by elimination. Finally, a new heuristic decoding method for linear codes over prime fields is discussed using Gr\"obner bases.