Coding theory package for Macaulay2

TL;DR

This paper introduces a Macaulay2 package that provides tools for constructing, analyzing, and decoding various linear codes using algebraic geometry and commutative algebra techniques.

Contribution

It implements a comprehensive set of functions for linear code parameters, important families, and decoding algorithms within Macaulay2, enhancing computational capabilities in coding theory.

Findings

Implemented functions for generator and parity check matrices.

Generated key families of linear codes like Reed--Solomon and Hamming.

Included decoding algorithms such as syndrome decoding.

Abstract

In this Macaulay2 \cite{M2} package we define an object called {\it linear code}. We implement functions that compute basic parameters and objects associated with a linear code, such as generator and parity check matrices, the dual code, length, dimension, and minimum distance, among others. We define an object {\it evaluation code}, a construction which allows to study linear codes using tools of algebraic geometry and commutative algebra. We implement functions to generate important families of linear codes such as Hamming codes, cyclic codes, Reed--Solomon codes, Reed--Muller codes, Cartesian codes, monomial--Cartesian codes, and toric codes. In addition, we define functions for the syndrome decoding algorithm and locally recoverable code construction, which are important tools in applications of linear codes. The package \textit{CodingTheory.m2} is available at \url{https://github.com/Macaulay2/Workshop-2020-Cleveland/tree/CodingTheory/CodingTheory}