Cyclic Codes As Submodules Of Rings And Direct Product Of Rings
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Author | : Roger Ten Valls |
Publisher | : |
Total Pages | : 128 |
Release | : 2017 |
Genre | : |
ISBN | : 9788449072062 |
Download Cyclic Codes as Submodules of Rings and Direct Product of Rings Book in PDF, Epub and Kindle
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.
Author | : S.K. Jain |
Publisher | : Oxford University Press |
Total Pages | : 231 |
Release | : 2012-09-27 |
Genre | : Mathematics |
ISBN | : 019966451X |
Download Cyclic Modules and the Structure of Rings Book in PDF, Epub and Kindle
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.
Author | : Serdar Boztas |
Publisher | : Springer |
Total Pages | : 379 |
Release | : 2007-11-29 |
Genre | : Computers |
ISBN | : 3540772243 |
Download Applied Algebra, Algebraic Algorithms and Error-Correcting Codes Book in PDF, Epub and Kindle
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.
Author | : Minjia Shi |
Publisher | : Academic Press |
Total Pages | : 320 |
Release | : 2017-06-12 |
Genre | : Mathematics |
ISBN | : 0128133910 |
Download Codes and Rings Book in PDF, Epub and Kindle
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
Author | : Patrick Sole |
Publisher | : World Scientific |
Total Pages | : 201 |
Release | : 2009-07-16 |
Genre | : Mathematics |
ISBN | : 9814468991 |
Download Codes Over Rings Book in PDF, Epub and Kindle
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.
Author | : M. Isaacs |
Publisher | : American Mathematical Soc. |
Total Pages | : 392 |
Release | : 1989 |
Genre | : Computers |
ISBN | : 0821850989 |
Download Representation Theory, Group Rings, and Coding Theory Book in PDF, Epub and Kindle
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.
Author | : Brett E. Leroux |
Publisher | : |
Total Pages | : 0 |
Release | : 2014 |
Genre | : Chinese remainder theorem |
ISBN | : |
Download Rings on the Direct Product of Two Cyclic Groups Book in PDF, Epub and Kindle
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.
Author | : |
Publisher | : |
Total Pages | : 534 |
Release | : 1995 |
Genre | : Mathematics |
ISBN | : |
Download Doklady Book in PDF, Epub and Kindle
Author | : Jonas Saman |
Publisher | : |
Total Pages | : 51 |
Release | : 2015 |
Genre | : Algebra |
ISBN | : |
Download Additive Cyclic Codes Over Rings Book in PDF, Epub and Kindle
"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 | : |
Publisher | : |
Total Pages | : 406 |
Release | : 2008 |
Genre | : Mathematics |
ISBN | : |
Download Applied Mathematics Book in PDF, Epub and Kindle