Algebraic Coding Theory Over Finite Commutative Rings

verfasst von: Steven T. Dougherty

Verlag: Springer International Publishing

Buchreihe : SpringerBriefs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book provides a self-contained introduction to algebraic coding theory over finite Frobenius rings. It is the first to offer a comprehensive account on the subject.

Coding theory has its origins in the engineering problem of effective electronic communication where the alphabet is generally the binary field. Since its inception, it has grown as a branch of mathematics, and has since been expanded to consider any finite field, and later also Frobenius rings, as its alphabet. This book presents a broad view of the subject as a branch of pure mathematics and relates major results to other fields, including combinatorics, number theory and ring theory.

Suitable for graduate students, the book will be of interest to anyone working in the field of coding theory, as well as algebraists and number theorists looking to apply coding theory to their own work.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In this chapter, we give a brief introduction to the history of algebraic coding theory from its inception in the 1940s to its reinvention as a branch of pure mathematics and give the basic denitions and notations necessary to begin a study of the subject.

Steven T. Dougherty

Chapter 2. Ring Theory

Abstract

We give the necessary definitions and foundational results from commutative ring theory for the study of codes over rings. We give the definition of Frobenius rings and characterize them in terms of characters. We prove the generalized Chinese Remainder Theorem and describe what constitutes a minimal generating set for a code over a finite Frobenius ring.

Steven T. Dougherty

Chapter 3. MacWilliams Relations

Abstract

In this Chapter, we prove the MacWilliams relations for codes over finite Frobenius commutative rings. These relations are one of the foundational results of algebraic coding theory. We describe them first for codes over groups and extend this to codes over Frobenius rings. Finally, we give a practical guide for producing MacWilliams relations for a specific ring.

Steven T. Dougherty

Chapter 4. Families of Rings

Abstract

In this chapter, we describe families of rings including the rings of order 4, their generalizations, X-rings, and the ring \(R_{q,\varDelta }\). We describe the kernel and rank of binary codes that are images of quaternary codes via the Gray map. Then a generalized Singleton bound is proven for codes over Frobenius rings.

Steven T. Dougherty

Chapter 5. Self-dual Codes

Abstract

In this chapter, we describe self-dual codes over Frobenius rings. We give constructions of self-dual codes over any Frobenius ring. We describe connections to unimodular lattices, binary self-dual codes and to designs. We also describe linear complementary dual codes and make a new definition of a broad generalization encompassing both self-dual and linear complementary codes.

Steven T. Dougherty

Chapter 6. Cyclic and Constacyclic Codes

Abstract

In this chapter, we study polycyclic, negacyclic, constacyclic, quasicyclic and skew cyclic codes which are all generalizations of the important family of cyclic codes. We describe their algebraic setting and show how to use this setting to classify these families of codes.

Steven T. Dougherty

Backmatter

Titel: Algebraic Coding Theory Over Finite Commutative Rings
verfasst von: Steven T. Dougherty
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-59806-2
Print ISBN: 978-3-319-59805-5
DOI: https://doi.org/10.1007/978-3-319-59806-2