Coding theory package for Macaulay2

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.1. INTRODUCTION. Coding theory has been extensively studied since 1948, when Claude Shannon [1948] proved in his seminal paper that linear codes can be used to reliably transmit information from a single source to a single receiver through a noisy channel. Since then, coding theory has found many important engineering applications. For example, coding theory has been used in designing reliable data storage systems, radio communication protocols, and in the emerging field of quantum computers. Coding theory has close ties with many areas in mathematics including linear algebra, commutative algebra, algebraic geometry, and combinatorics.In this note we introduce the new [Macaulay2] package called CodingTheory. The goal of this package is to provide a range of functions for constructing linear and evaluation codes over finite fields, and for computing some of their main properties. To this aim, we implement two types of objects, LinearCode and EvaluationCode. The package also includes implementations of functions for generating important families of linear codes like Hamming codes, cyclic codes, Reed-Solomon codes, Reed-Muller codes, Cartesian codes, monomial-Cartesian codes and toric codes. It also has functions for the syndrome decoding algorithm and locally recoverable codes.The organization of this note is as follows. In Section 2 we describe various ways to construct a linear code over a finite field using the CodingTheory package. In Section 3 we show how to compute the main parameters of a linear code: length, dimension, and minimum distance. We also illustrate how to

The dual of an evaluation code

“…Let L be the linear space generated by B = {1, t 1 , t 2 , t 1 t 2 } and let L T be the monomial standard evaluation code on T = A 1 × A 2 relative to the GRevLex order ≺. The vanishing ideal I = I(T ) of T is generated by t 3 1 − 1 and t 2 2 − 1, the index of regularity of H a I is 3, and the set of standard monomials of S/I is ∆ ≺ (I) = {1, t 1 , t 2 , t 2 1 , t 1 t 2 , t 2 1 t 2 }. According to Proposition 7.2 and Corollary 7.3, the algebraic dual L ⊥ is given by…”