Universal Gr\"obner Bases for Binary Linear Codes

TL;DR

This paper explores the algebraic structure of binary linear codes through code ideals, establishing relationships between various bases and introducing a new class called Singleton codes based on Gr"obner basis properties.

Contribution

It extends toric ideal concepts to code ideals, characterizes the universal Gr"obner basis for binary codes, and introduces Singleton codes with specific algebraic properties.

Findings

Universal Gr"obner basis includes codewords satisfying the Singleton bound and a rank condition.

Established inclusion relations between circuits, Gr"obner basis, and Graver basis for code ideals.

Introduced Singleton codes as a new class of binary linear codes.

Abstract

Each linear code can be described by a code ideal given as the sum of a toric ideal and a non-prime ideal. In this way, several concepts from the theory of toric ideals can be translated into the setting of code ideals. It will be shown that after adjusting some of these concepts, the same inclusion relationship between the set of circuits, the universal Gr\"obner basis and the Graver basis holds. Furthermore, in the case of binary linear codes, the universal Gr\"obner basis will consist of all binomials which correspond to codewords that satisfy the Singleton bound and a particular rank condition. This will give rise to a new class of binary linear codes denoted as Singleton codes.