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, decomposition theory of quasi-cyclic codes and related codes and duality over Frobenius rings. Primarily suitable for ring theorists at PhD level engaged in application research and coding theorists interested in algebraic foundations, the work is also valuable to computational scientists and working cryptologists in the area. - Consolidates 20+ years of research in one volume, helping researchers save time in the evaluation of disparate literature - Discusses duality formulas in the context of Frobenius rings - Reviews decomposition of quasi-cyclic codes under ring action - Evaluates the ideal and modular structure of skew-cyclic codes - Supports applications in data compression, distributed storage, network coding, cryptography and across error-correction

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.

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.

The12thintheseriesofIMAConferencesonCryptographyandCodingwasheld at the Royal Agricultural College, Cirencester, December 15–17, 2009. The p- gram comprised 3 invited talks and 26 contributed talks. The contributed talks werechosenbyathoroughreviewingprocessfrom53submissions.Oftheinvited and contributed talks,28 arerepresentedaspapersin this volume. These papers are grouped loosely under the headings: Coding Theory, Symmetric Crypt- raphy, Security Protocols, Asymmetric Cryptography, Boolean Functions, and Side Channels and Implementations. Numerous people helped to make this conference a success. To begin with I would like to thank all members of the Technical Program Committee who put a great deal of e?ort into the reviewing process so as to ensure a hi- quality program. Moreover, I wish to thank a number of people, external to the committee, who also contributed reviews on the submitted papers. Thanks, of course,mustalso goto allauthorswho submitted papers to the conference,both those rejected and accepted. The review process was also greatly facilitated by the use of the Web-submission-and-review software, written by Shai Halevi of IBM Research, and I would like to thank him for making this package available to the community. The invited talks were given by Frank Kschischang, Ronald Cramer, and Alexander Pott, and two of these invitedtalksappearaspapersinthisvolume. A particular thanks goes to these invited speakers, each of whom is well-known, notonlyforbeingaworld-leaderintheir?eld,butalsofortheirparticularability to communicate their expertise in an enjoyable and stimulating manner.

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 such as Construction A or Reed–Muller codes. Classical topics like enveloping skewfields, weak Hopf algebras, and tropical algebras are also presented.

Algebraic geometry is introduced, with particular attention given to projective curves, rational functions and divisors. The construction of algebraic geometric codes is given, and the Tsfasman-Vladut-Zink result mentioned above it discussed."--BOOK JACKET.

One of the most remarkable and beautiful theorems in coding theory is Gleason's 1970 theorem about the weight enumerators of self-dual codes and their connections with invariant theory, which has inspired hundreds of papers about generalizations and applications of this theorem to different types of codes. This self-contained book develops a new theory which is powerful enough to include all the earlier generalizations.