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
Abstract. Because of its interesting applications in coding theory, cryptography, and algebraic combinatorics, in recent decades a lot of attention has been paid to the algebraic structure of the ring of polynomials R [x], where R is a finite commutative ring with identity. Motivated by this popularity, in this paper we determine when R[x] is a principal ideal ring. In fact, we prove that R[x] is a principal ideal ring if and only if R is a finite direct product of finite fields. Keywords: Principal ideal ring, polynomial ring, finite rings. MSC2010: 13F10, 13F20, 16P10, 13C05. ¿Cuándo R[x] es un anillo de ideales principales?Resumen. Debido a sus interesantes aplicaciones en teoría de códigos, criptografía y combinatoria algebraica, en décadas recientes se ha incrementado la atención en la estructura algebraica del anillo de polinomios R[x], donde R es un anillo conmutativo finito con identidad. Motivados por esta popularidad, en este artículo determinamos cuándo R[x] es un anillo de ideales principales. De hecho, demostramos que R[x] es un anillo de ideales principales, si y sólo si, R es un producto directo finito de campos finitos. Palabras clave: Anillo de ideales principales, anillo de polinomios, anillos finitos.
It has recently been shown that a minimal reversible nonsymmetric ring has order 256 answering a questioned original posed in a paper on a taxonomy of 2-primal rings. Answers to similar questions on minimal rings relating to this taxonomy were also answered in a related work. One type of minimal ring that was left out of that report, was a minimal abelian reflexive nonsemicommutative ring. In this work it is shown that a minimal abelian reflexive nonsemicommutative ring is of order 256 an example of which is F2D8. This is a consequence of the other primary result which is that a finite abelian reflexive ring of order p k for some prime p and k < 8 is reversible.
Circulant matrices are an important tool widely used in coding theory and cryptography. A circulant matrix is a square matrix whose rows are the cyclic shifts of the first row. Such a matrix can be efficiently stored in memory because it is fully specified by its first row. The ring of n × n circulant matrices can be identified with the quotient ring F[x]/(x n − 1). In consequence, the strong algebraic structure of the ring F[x]/(x n − 1) can be used to study properties of the collection of all n × n circulant matrices. The ring F[x]/(x n − 1) is a special case of a group algebra and elements of any finite dimensional group algebra can be represented with square matrices which are specified by a single column. In this paper we study this representation and prove that it is an injective Hamming weight preserving homomorphism of F-algebras and classify it in the case where the underlying group is abelian.Our work is motivated by the desire to generalize the BIKE cryptosystem (a contender in the NIST competition to get a new postquantum standard for asymmetric cryptography). Group algebras can be used to design similar cryptosystems or, more generally, to construct low density or moderate density parity-check matrices for linear codes.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.