Codes and Rings

• Author : Minjia Shi,Adel Alahmadi,Patrick Solé
• Publisher : Academic Press
• Release : 12 June 2017

Codes and Rings: Theory and Practice is a systematic review of literature that focuses on codes over rings and rings acting on codes. Since the breakthrough works on quaternary codes in the 1990s, two decades of research have moved the field far beyond its original periphery. This book fills this gap by consolidating results scattered in the literature, addressing classical as well as applied aspects of rings and coding theory. New research covered by the book encompasses skew cyclic codes,

• Author : Anonim
• Publisher : Unknown
• Release : 03 August 2021

• Author : Ian F. Blake,Ronald C. Mullin
• Publisher : Academic Press
• Release : 10 May 2014

The Mathematical Theory of Coding focuses on the application of algebraic and combinatoric methods to the coding theory, including linear transformations, vector spaces, and combinatorics. The publication first offers information on finite fields and coding theory and combinatorial constructions and coding. Discussions focus on self-dual and quasicyclic codes, quadratic residues and codes, balanced incomplete block designs and codes, bounds on code dictionaries, code invariance under permutation groups, and linear transformations of vector spaces over finite fields. The text then takes

• Author : Steven T. Dougherty
• Publisher : Springer
• Release : 04 July 2017

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

• Author : Roger Ten Valls
• Publisher : Unknown
• Release : 03 August 2021

Coding theory originated as the mathematical foundation for the transmission of messages over noisy communication channels and deals with the problem of detecting and correcting transmission errors caused by the noise of the channel. The mathematical background needed for example was linear algebra, theory of groups, rings and nite elds, and other areas of discrete mathematics, as theory of designs. Thus, coding theory has now become and active part of mathematical research. On the family of codes, linear codes are

• Author : M. Isaacs
• Publisher : American Mathematical Soc.
• Release : 03 August 1989

This volume is dedicated to the memory of the Soviet mathematician S. D. Berman (1922-1987). Berman's work - for the most part in representation theory, group rings, and coding theory - is discussed here in a number of review articles. Among the topics covered are Berman's achievements in coding theory, including his pioneering work on abelian codes and his results on the theory of threshold functions. Also discussed are his contributions to the representation theory of groups over fields, his

• Author : André Leroy,Christian Lomp,Sergio López-Permouth,Frédérique Oggier
• Publisher : American Mathematical Soc.
• Release : 12 April 2019

This book contains the proceedings of the Fifth International Conference on Noncommutative Rings and their Applications, held from June 12–15, 2017, at the University of Artois, Lens, France. The papers are related to noncommutative rings, covering topics such as: ring theory, with both the elementwise and more structural approaches developed; module theory with popular topics such as automorphism invariance, almost injectivity, ADS, and extending modules; and coding theory, both the theoretical aspects such as the extension theorem and the more applied ones

• Author : Judy Leavitt Walker
• Publisher : Unknown
• Release : 03 August 1996

The techniques of algebraic geometry have been widely and successfully applied to the study of linear codes over finite fields since the early 1980's. Recently, there has been an increased interest in the study of linear codes over finite rings. In this thesis, we combine these two approaches to coding theory by introducing and studying algebraic geometric codes over rings. We define algebraic geometric codes over any local Artinian ring A and compute their parameters. Under the additional hypothesis that

• Author : Patrick Solé
• Publisher : World Scientific Publishing Company Incorporated
• Release : 03 August 2021

This is the proceedings volume of the International Centre for Pure and Applied Mathematics Summer School course held in Ankara, Turkey, in August 2008. Contributors include Greferath, Honold, Landgev, Ling, Lopez, Nebe, Nechaev, Özbudak, Solé, Wolfmann and Wood. The aim is to present a survey in fundamental areas and highlight some recent results.

• Author : Minjia Shi,Youngju Choie,Anuradha Sharma,Patrick Sole
• Publisher : World Scientific
• Release : 20 November 2019

There are connections between invariant theory and modular forms since the times of Felix Klein, in the 19th century, connections between codes and lattices since the 1960's. The aim of the book is to explore the interplay between codes and modular forms. Here modular form is understood in a wide sense (Jacobi forms, Siegel forms, Hilbert forms). Codes comprises not only linear spaces over finite fields but modules over some commutative rings. The connection between codes over finite fields and

• Author : Gilberto Bini,Flaminio Flamini
• Publisher : Springer Science & Business Media
• Release : 06 December 2012

Foreword by Dieter Jungnickel Finite Commutative Rings and their Applications answers a need for an introductory reference in finite commutative ring theory as applied to information and communication theory. This book will be of interest to both professional and academic researchers in the fields of communication and coding theory. The book is a concrete and self-contained introduction to finite commutative local rings, focusing in particular on Galois and Quasi-Galois rings. The reader is provided with an active and concrete approach

• Author : Isaac Woungang,Sudip Misra,Subhas Chandra Misra
• Publisher : World Scientific
• Release : 03 August 2021

The last few years have witnessed rapid advancements in information and coding theory research and applications. This book provides a comprehensive guide to selected topics, both ongoing and emerging, in information and coding theory. Consisting of contributions from well-known and high-profile researchers in their respective specialties, topics that are covered include source coding; channel capacity; linear complexity; code construction, existence and analysis; bounds on codes and designs; space-time coding; LDPC codes; and, codes and cryptography. All of the chapters are

• Author : Steven Dougherty,Alberto Facchini,Andre Gerard Leroy,Edmund Puczylowski,Patrick Sole
• Publisher : American Mathematical Soc.
• Release : 20 February 2015

This volume contains the Proceedings of an International Conference on Noncommutative Rings and Their Applications, held July 1-4, 2013, at the Universite d'Artois, Lens, France. It presents recent developments in the theories of noncommutative rings and modules over such rings as well as applications of these to coding theory, enveloping algebras, and Leavitt path algebras. Material from the course ``Foundations of Algebraic Coding Theory``, given by Steven Dougherty, is included and provides the reader with the history and background of coding

• Author : Jonas Saman
• Publisher : Unknown
• Release : 03 August 2021

"This Master's thesis introduces a reader with an average knowledge of mathematics to coding theory. A background in algebra is included with the required details needed in coding theory. The thesis guides the reader from casual examples to linear codes, as well as cyclic codes. The focus lies on Z2Z4-additive cyclic codes. At the end new theorems regarding Z2Z4-additive cyclic codes with even in Z2[x]=(x {u100000} 1) Z4[x]=(x {u100000} 1) are discussed."--Abstract.

• Author : Gregory Reid Holdman
• Publisher : Unknown
• Release : 03 August 2021

The theory of error-correcting codes has historically been most useful in the context of linear codes. Such codes may be viewed as vector spaces over Galois fields carrying with them many familiar and well-studied properties. A generalization of Galois fields is the concept of Galois rings. It is therefore natural to consider codes over Galois rings to study which properties such codes maintain in the move to a more general setting. This thesis will present two separate expositions on coding