Vol. 11, No. 1, 2021

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Coding theory package for Macaulay2

Taylor Ball, Eduardo Camps, Henry Chimal-Dzul, Delio Jaramillo-Velez, Hiram López, Nathan Nichols, Matthew Perkins, Ivan Soprunov, German Vera-Martínez and Gwyn Whieldon

Vol. 11 (2021), 113–122
Abstract

In this Macaulay2 package we implement a type of object called a LinearCode. 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 implement a type of object called an EvaluationCode, a construction which allows users 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 implement functions for the syndrome decoding algorithm and locally recoverable code construction, which are important tools in applications of linear codes.

Keywords
linear codes, locally recoverable codes, Cartesian codes, evaluation codes, Hamming codes
Mathematical Subject Classification
Primary: 13P25, 94B05
Secondary: 11T71, 14G50
Supplementary material

Coding theory and linear codes computations

Milestones
Received: 13 July 2020
Revised: 16 July 2021
Accepted: 10 August 2021
Published: 20 February 2022
Authors
Taylor Ball
University of Notre Dame
Notre Dame, IN
United States
Eduardo Camps
Escuela Superior de Física y Matemáticas
Instituto Politecnico Nacional
Zacatenco
Mexico City
Mexico
Henry Chimal-Dzul
Department of Mathematics and Center of Ring Theory and its Applications
Ohio University
Athens, OH
United States
Delio Jaramillo-Velez
Departamento de Matemáticas
Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional
Mexico City
Mexico
Hiram López
Department of Mathematics and Statistics
Cleveland State University
Cleveland, OH
United States
Nathan Nichols
School of Mathematics
University of Minnesota Twin Cities
Minneapolis, MN
United States
Matthew Perkins
Department of Mathematics and Statistics
Cleveland State University
Cleveland, OH
United States
Ivan Soprunov
Department of Mathematics and Statistics
Cleveland State University
Cleveland, OH
United States
German Vera-Martínez
Escuela Superior de Física y Matemáticas
Instituto Politecnico Nacional
Zacatenco
Mexico City
Mexico
Gwyn Whieldon
Frederick, MD
United States