Cyclic Codes as Submodules of Rings and Direct Product of Rings

Cyclic Codes as Submodules of Rings and Direct Product of Rings
Author: Roger Ten Valls
Publisher:
Total Pages: 128
Release: 2017
Genre:
ISBN: 9788449072062

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Els codis cíclics són una família important en la teoria de la codificació i han estat una àrea principal d'estudi des de la seva aparició. Fins a la dècada dels 90, els alfabets habitualment utilitzats en teoria de codis eren cossos finits. A partir d'aleshores, es va iniciar l'estudi de codis definits sobre anells. Des de l'aparició dels codis Z2Z4-additius, la investigació de codis sobre alfabets d'anells mixtes s'ha incrementat. L'any 2014, Abualrub et al. van presentar els codis cíclics Z2Z4-additius i aquest fet va marcar l'inici de l'estudi de les propietats cícliques en codis sobre alfabets mixtes. La present tesi té com a objectiu examinar l'estructura algebraica dels codis cíclics com a submòduls de productes directes d'anells finits. Partint del fet que aquests codis poden ser interpretats com a submòduls del producte directe d'anells de polinomis, es determina l'estructura d'aquests codis cíclics tot donant els seus polinomis generadors. A més, s'estudia el concepte de dualitat definint l'operació polinòmica corresponent al producte intern de vectors. Aquesta operació permet entendre la dualitat en l'anell de polinomis corresponent. Així mateix, es proporcionen tècniques per donar una descripció polinòmica dels codis duals en termes dels polinomis generadors dels codis cíclics i es calculen explícitament en alguns casos particulars. També es consideren diferents mètriques en el producte directe d'anells finits i s'estudien les seves imatges binàries a través de diferents aplicacions que preserven les distàncies, anomenades aplicacions de Gray. Finalment, es dóna una estructura algebraica per a una gran família de codis quasi-cíclics binaris tot construint una família d'anells commutatius i una aplicació de Gray canònica. De tal manera que els codis cíclics sobre aquesta família d'anells produeixen codis quasi-cíclics d'índex arbitrari en l'espai d'Hamming a través de l'aplicació de Gray.

Cyclic Modules and the Structure of Rings

Cyclic Modules and the Structure of Rings
Author: S.K. Jain
Publisher: Oxford University Press
Total Pages: 231
Release: 2012-09-27
Genre: Mathematics
ISBN: 019966451X

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This unique monograph brings together important material in the field of noncommutative rings and modules. It provides an up-to-date account of the topic of cyclic modules and the structure of rings which will be of particular interest to those working in abstract algebra and to graduate students who are exploring potential research topics.

Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Author: Serdar Boztas
Publisher: Springer
Total Pages: 379
Release: 2007-11-29
Genre: Computers
ISBN: 3540772243

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This book constitutes the refereed proceedings of the 17th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-17, held in Bangalore, India, in December 2007. Among the subjects addressed are block codes, including list-decoding algorithms; algebra and codes: rings, fields, algebraic geometry codes; algebra: rings and fields, polynomials, permutations, lattices; cryptography: cryptanalysis and complexity; computational algebra.

Codes and Rings

Codes and Rings
Author: Minjia Shi
Publisher: Academic Press
Total Pages: 320
Release: 2017-06-12
Genre: Mathematics
ISBN: 0128133910

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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

Codes Over Rings

Codes Over Rings
Author: Patrick Sole
Publisher: World Scientific
Total Pages: 201
Release: 2009-07-16
Genre: Mathematics
ISBN: 9814468991

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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 Boztaş, Udaya, Dinh, Ling, López-Permouth, Szabo, Honold, Landjev and Wood. The aim is to present a survey in fundamental areas and highlight some recent results.

Representation Theory, Group Rings, and Coding Theory

Representation Theory, Group Rings, and Coding Theory
Author: M. Isaacs
Publisher: American Mathematical Soc.
Total Pages: 392
Release: 1989
Genre: Computers
ISBN: 0821850989

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Dedicated to the memory of the Soviet mathematician S D Berman (1922-1987), this work covers topics including Berman's achievements in coding theory, including his pioneering work on abelian codes and his results on the theory of threshold functions.

Rings on the Direct Product of Two Cyclic Groups

Rings on the Direct Product of Two Cyclic Groups
Author: Brett E. Leroux
Publisher:
Total Pages: 0
Release: 2014
Genre: Chinese remainder theorem
ISBN:

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The focus of this paper is a classification of rings whose additive group is the direct product of two cyclic groups. Such rings are represented by a quotient ring of the polynomials with integer coefficients. The paper begins with an overview of general ring theory including the Chinese Remainder theorem and the theory of local/irreducible rings. We then introduce Hensel's lemma which is later used as the main tool for classifying rings on the direct product of two cyclic groups. It is shown that two of these rings are isomorphic if and only if there is a solution to a particular quadratic equation in two variables mod n. We derive a new form of Hensel's lemma that applies directly to quadratic equations in two variables. It is used to systematically solve the quadratics in question and thus obtain a complete classification of rings on the direct product of two cyclic groups.

Doklady

Doklady
Author:
Publisher:
Total Pages: 534
Release: 1995
Genre: Mathematics
ISBN:

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Additive Cyclic Codes Over Rings

Additive Cyclic Codes Over Rings
Author: Jonas Saman
Publisher:
Total Pages: 51
Release: 2015
Genre: Algebra
ISBN:

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"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.

Applied Mathematics

Applied Mathematics
Author:
Publisher:
Total Pages: 406
Release: 2008
Genre: Mathematics
ISBN:

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