Algebraic Geometry Codes Advanced Chapters

Author : Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin
Publisher : American Mathematical Soc.
Page : 453 pages
File Size : 41,8 Mb
Release : 2019-07-02
Category : Coding theory
ISBN : 9781470448653

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Algebraic Geometry Codes: Advanced Chapters by Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin Pdf

Algebraic Geometry Codes: Advanced Chapters is devoted to the theory of algebraic geometry codes, a subject related to local_libraryBook Catalogseveral domains of mathematics. On one hand, it involves such classical areas as algebraic geometry and number theory; on the other, it is connected to information transmission theory, combinatorics, finite geometries, dense packings, and so on. The book gives a unique perspective on the subject. Whereas most books on coding theory start with elementary concepts and then develop them in the framework of coding theory itself within, this book systematically presents meaningful and important connections of coding theory with algebraic geometry and number theory. Among many topics treated in the book, the following should be mentioned: curves with many points over finite fields, class field theory, asymptotic theory of global fields, decoding, sphere packing, codes from multi-dimensional varieties, and applications of algebraic geometry codes. The book is the natural continuation of Algebraic Geometric Codes: Basic Notions by the same authors. The concise exposition of the first volume is included as an appendix.

Author : Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin
Publisher : American Mathematical Society
Page : 338 pages
File Size : 52,9 Mb
Release : 2022-04-15
Category : Mathematics
ISBN : 9781470470074

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Algebraic Geometric Codes: Basic Notions by Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin Pdf

The book is devoted to the theory of algebraic geometric codes, a subject formed on the border of several domains of mathematics. On one side there are such classical areas as algebraic geometry and number theory; on the other, information transmission theory, combinatorics, finite geometries, dense packings, etc. The authors give a unique perspective on the subject. Whereas most books on coding theory build up coding theory from within, starting from elementary concepts and almost always finishing without reaching a certain depth, this book constantly looks for interpretations that connect coding theory to algebraic geometry and number theory. There are no prerequisites other than a standard algebra graduate course. The first two chapters of the book can serve as an introduction to coding theory and algebraic geometry respectively. Special attention is given to the geometry of curves over finite fields in the third chapter. Finally, in the last chapter the authors explain relations between all of these: the theory of algebraic geometric codes.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 44,7 Mb
Release : 2024-06-10
Category : Electronic
ISBN : 9789814471619

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Advances in Algebraic Geometry Codes by Anonim Pdf

Author : Wolfgang Ebeling
Publisher : Springer Science & Business Media
Page : 167 pages
File Size : 42,6 Mb
Release : 2012-09-18
Category : Mathematics
ISBN : 9783658003609

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Lattices and Codes by Wolfgang Ebeling Pdf

The purpose of coding theory is the design of efficient systems for the transmission of information. The mathematical treatment leads to certain finite structures: the error-correcting codes. Surprisingly problems which are interesting for the design of codes turn out to be closely related to problems studied partly earlier and independently in pure mathematics. In this book, examples of such connections are presented. The relation between lattices studied in number theory and geometry and error-correcting codes is discussed. The book provides at the same time an introduction to the theory of integral lattices and modular forms and to coding theory. In the 3rd edition, again numerous corrections and improvements have been made and the text has been updated.

Author : Judy L. Walker
Publisher : American Mathematical Soc.
Page : 82 pages
File Size : 40,8 Mb
Release : 2000
Category : Coding theory
ISBN : 9780821826287

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Codes and Curves by Judy L. Walker Pdf

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.

Author : Richard E. Blahut
Publisher : Cambridge University Press
Page : 10 pages
File Size : 48,9 Mb
Release : 2008-04-03
Category : Technology & Engineering
ISBN : 9781139469463

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Algebraic Codes on Lines, Planes, and Curves by Richard E. Blahut Pdf

The past few years have witnessed significant developments in algebraic coding theory. This book provides an advanced treatment of the subject from an engineering perspective, covering the basic principles and their application in communications and signal processing. Emphasis is on codes defined on the line, on the plane, and on curves, with the core ideas presented using commutative algebra and computational algebraic geometry made accessible using the Fourier transform. Starting with codes defined on a line, a background framework is established upon which the later chapters concerning codes on planes, and on curves, are developed. The decoding algorithms are developed using the standard engineering approach applied to those of Reed-Solomon codes, enabling them to be evaluated against practical applications. Integrating recent developments in the field into the classical treatment of algebraic coding, this is an invaluable resource for graduate students and researchers in telecommunications and applied mathematics.

Author : Henning Stichtenoth
Publisher : Springer Science & Business Media
Page : 360 pages
File Size : 42,8 Mb
Release : 2009-02-11
Category : Mathematics
ISBN : 9783540768784

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Algebraic Function Fields and Codes by Henning Stichtenoth Pdf

This book links two subjects: algebraic geometry and coding theory. It uses a novel approach based on the theory of algebraic function fields. Coverage includes the Riemann-Rock theorem, zeta functions and Hasse-Weil's theorem as well as Goppa' s algebraic-geometric codes and other traditional codes. It will be useful to researchers in algebraic geometry and coding theory and computer scientists and engineers in information transmission.

Author : Simeon Ball
Publisher : Springer Nature
Page : 185 pages
File Size : 46,5 Mb
Release : 2020-05-08
Category : Mathematics
ISBN : 9783030411534

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A Course in Algebraic Error-Correcting Codes by Simeon Ball Pdf

This textbook provides a rigorous mathematical perspective on error-correcting codes, starting with the basics and progressing through to the state-of-the-art. Algebraic, combinatorial, and geometric approaches to coding theory are adopted with the aim of highlighting how coding can have an important real-world impact. Because it carefully balances both theory and applications, this book will be an indispensable resource for readers seeking a timely treatment of error-correcting codes. Early chapters cover fundamental concepts, introducing Shannon’s theorem, asymptotically good codes and linear codes. The book then goes on to cover other types of codes including chapters on cyclic codes, maximum distance separable codes, LDPC codes, p-adic codes, amongst others. Those undertaking independent study will appreciate the helpful exercises with selected solutions. A Course in Algebraic Error-Correcting Codes suits an interdisciplinary audience at the Masters level, including students of mathematics, engineering, physics, and computer science. Advanced undergraduates will find this a useful resource as well. An understanding of linear algebra is assumed.

Author : Oliver Pretzel
Publisher : Clarendon Press
Page : 209 pages
File Size : 44,7 Mb
Release : 1998-01-08
Category : Mathematics
ISBN : 9780191589041

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Codes and Algebraic Curves by Oliver Pretzel Pdf

The geometry of curves has fascinated mathematicians for 2500 years, and the theory has become highly abstract. Recently links have been made with the subject of error correction, leading to the creation of geometric Goppa codes, a new and important area of coding theory. This book is an updated and extended version of the last part of the successful book Error-Correcting Codes and Finite Fields. It provides an elementary introduction to Goppa codes, and includes many examples, calculations, and applications. The book is in two parts with an emphasis on motivation, and applications of the theory take precedence over proofs of theorems. The formal theory is, however, provided in the second part of the book, and several of the concepts and proofs have been simplified without sacrificing rigour.

Author : Dirk Hachenberger,Dieter Jungnickel
Publisher : Springer Nature
Page : 785 pages
File Size : 42,6 Mb
Release : 2020-09-29
Category : Mathematics
ISBN : 9783030608064

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Topics in Galois Fields by Dirk Hachenberger,Dieter Jungnickel Pdf

This monograph provides a self-contained presentation of the foundations of finite fields, including a detailed treatment of their algebraic closures. It also covers important advanced topics which are not yet found in textbooks: the primitive normal basis theorem, the existence of primitive elements in affine hyperplanes, and the Niederreiter method for factoring polynomials over finite fields. We give streamlined and/or clearer proofs for many fundamental results and treat some classical material in an innovative manner. In particular, we emphasize the interplay between arithmetical and structural results, and we introduce Berlekamp algebras in a novel way which provides a deeper understanding of Berlekamp's celebrated factorization algorithm. The book provides a thorough grounding in finite field theory for graduate students and researchers in mathematics. In view of its emphasis on applicable and computational aspects, it is also useful for readers working in information and communication engineering, for instance, in signal processing, coding theory, cryptography or computer science.

Author : Ruud Pellikaan,Xin-Wen Wu,Stanislav Bulygin,Relinde Jurrius
Publisher : Cambridge University Press
Page : 611 pages
File Size : 43,5 Mb
Release : 2017-11-02
Category : Language Arts & Disciplines
ISBN : 9780521817110

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Codes, Cryptology and Curves with Computer Algebra by Ruud Pellikaan,Xin-Wen Wu,Stanislav Bulygin,Relinde Jurrius Pdf

Graduate-level introduction to error-correcting codes, which are used to protect digital data and applied in public key cryptosystems.

Author : Paul B. Larson,Jindrich Zapletal
Publisher : American Mathematical Soc.
Page : 330 pages
File Size : 55,7 Mb
Release : 2020-07-16
Category : Education
ISBN : 9781470454623

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Geometric Set Theory by Paul B. Larson,Jindrich Zapletal Pdf

This book introduces a new research direction in set theory: the study of models of set theory with respect to their extensional overlap or disagreement. In Part I, the method is applied to isolate new distinctions between Borel equivalence relations. Part II contains applications to independence results in Zermelo–Fraenkel set theory without Axiom of Choice. The method makes it possible to classify in great detail various paradoxical objects obtained using the Axiom of Choice; the classifying criterion is a ZF-provable implication between the existence of such objects. The book considers a broad spectrum of objects from analysis, algebra, and combinatorics: ultrafilters, Hamel bases, transcendence bases, colorings of Borel graphs, discontinuous homomorphisms between Polish groups, and many more. The topic is nearly inexhaustible in its variety, and many directions invite further investigation.

Author : J. Scott Carter,Seiichi Kamada
Publisher : American Mathematical Society
Page : 365 pages
File Size : 42,9 Mb
Release : 2021-12-15
Category : Mathematics
ISBN : 9781470466718

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Diagrammatic Algebra by J. Scott Carter,Seiichi Kamada Pdf

This book is an introduction to techniques and results in diagrammatic algebra. It starts with abstract tensors and their categorifications, presents diagrammatic methods for studying Frobenius and Hopf algebras, and discusses their relations with topological quantum field theory and knot theory. The text is replete with figures, diagrams, and suggestive typography that allows the reader a glimpse into many higher dimensional processes. The penultimate chapter summarizes the previous material by demonstrating how to braid 3- and 4- dimensional manifolds into 5- and 6-dimensional spaces. The book is accessible to post-qualifier graduate students, and will also be of interest to algebraists, topologists and algebraic topologists who would like to incorporate diagrammatic techniques into their research.

Author : Nabil H. Mustafa
Publisher : American Mathematical Society
Page : 251 pages
File Size : 46,9 Mb
Release : 2022-01-14
Category : Mathematics
ISBN : 9781470461560

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Sampling in Combinatorial and Geometric Set Systems by Nabil H. Mustafa Pdf

Understanding the behavior of basic sampling techniques and intrinsic geometric attributes of data is an invaluable skill that is in high demand for both graduate students and researchers in mathematics, machine learning, and theoretical computer science. The last ten years have seen significant progress in this area, with many open problems having been resolved during this time. These include optimal lower bounds for epsilon-nets for many geometric set systems, the use of shallow-cell complexity to unify proofs, simpler and more efficient algorithms, and the use of epsilon-approximations for construction of coresets, to name a few. This book presents a thorough treatment of these probabilistic, combinatorial, and geometric methods, as well as their combinatorial and algorithmic applications. It also revisits classical results, but with new and more elegant proofs. While mathematical maturity will certainly help in appreciating the ideas presented here, only a basic familiarity with discrete mathematics, probability, and combinatorics is required to understand the material.

Author : Dana P. Williams
Publisher : American Mathematical Soc.
Page : 398 pages
File Size : 53,9 Mb
Release : 2019-09-24
Category : C*-algebras
ISBN : 9781470451332

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Tool Kit for Groupoid C∗ -Algebras by Dana P. Williams Pdf

The construction of a C∗-algebra from a locally compact groupoid is an important generalization of the group C∗-algebra construction and of the transformation group C∗-algebra construction. Since their introduction in 1980, groupoid C∗-algebras have been intensively studied with diverse applications, including graph algebras, classification theory, variations on the Baum-Connes conjecture, and noncommutative geometry. This book provides a detailed introduction to this vast subject and is suitable for graduate students or any researcher who wants to use groupoid C∗-algebras in their work. The main focus is to equip the reader with modern versions of the basic technical tools used in the subject, which will allow the reader to understand fundamental results and make contributions to various areas in the subject. Thus, in addition to covering the basic properties and construction of groupoid C∗-algebras, the focus is to give a modern treatment of some of the major developments in the subject in recent years, including the Equivalence Theorem and the Disintegration Theorem. Also covered are the complicated subjects of amenability of groupoids and simplicity results. The book is reasonably self-contained and accessible to graduate students with a good background in operator algebras.