Codes On Algebraic Curves

Author : Serguei A. Stepanov
Publisher : Springer Science & Business Media
Page : 352 pages
File Size : 40,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461547853

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Codes on Algebraic Curves by Serguei A. Stepanov Pdf

This is a self-contained introduction to algebraic curves over finite fields and geometric Goppa codes. There are four main divisions in the book. The first is a brief exposition of basic concepts and facts of the theory of error-correcting codes (Part I). The second is a complete presentation of the theory of algebraic curves, especially the curves defined over finite fields (Part II). The third is a detailed description of the theory of classical modular curves and their reduction modulo a prime number (Part III). The fourth (and basic) is the construction of geometric Goppa codes and the production of asymptotically good linear codes coming from algebraic curves over finite fields (Part IV). The theory of geometric Goppa codes is a fascinating topic where two extremes meet: the highly abstract and deep theory of algebraic (specifically modular) curves over finite fields and the very concrete problems in the engineering of information transmission. At the present time there are two essentially different ways to produce asymptotically good codes coming from algebraic curves over a finite field with an extremely large number of rational points. The first way, developed by M. A. Tsfasman, S. G. Vladut and Th. Zink [210], is rather difficult and assumes a serious acquaintance with the theory of modular curves and their reduction modulo a prime number. The second way, proposed recently by A.

Author : Oliver Pretzel
Publisher : Clarendon Press
Page : 209 pages
File Size : 52,8 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 : Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin
Publisher : American Mathematical Soc.
Page : 453 pages
File Size : 47,9 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 : Judy L. Walker
Publisher : American Mathematical Soc.
Page : 82 pages
File Size : 52,6 Mb
Release : 2000
Category : Computers
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 : Henning Stichtenoth
Publisher : Springer Science & Business Media
Page : 360 pages
File Size : 41,5 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 : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 50,9 Mb
Release : 2024-07-01
Category : Electronic
ISBN : 9789814471619

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Advances in Algebraic Geometry Codes by Anonim Pdf

Author : M. Tsfasman,S.G. Vladut
Publisher : Springer Science & Business Media
Page : 671 pages
File Size : 47,8 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9789401138109

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Algebraic-Geometric Codes by M. Tsfasman,S.G. Vladut Pdf

'Et moi ..., si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point aIle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d' etre of this series.

Author : Edgar Martinez-Moro
Publisher : World Scientific
Page : 453 pages
File Size : 45,5 Mb
Release : 2008
Category : Computers
ISBN : 9789812794017

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Advances in Algebraic Geometry Codes by Edgar Martinez-Moro Pdf

Advances in Algebraic Geometry Codes presents the most successful applications of algebraic geometry to the field of error-correcting codes, which are used in the industry when one sends information through a noisy channel. The noise in a channel is the corruption of a part of the information due to either interferences in the telecommunications or degradation of the information-storing support (for instance, compact disc). An error-correcting code thus adds extra information to the message to be transmitted with the aim of recovering the sent information. With contributions from renowned researchers, this pioneering book will be of value to mathematicians, computer scientists, and engineers in information theory.

Author : Goppa
Publisher : Springer Science & Business Media
Page : 166 pages
File Size : 46,9 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9789401568708

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Geometry and Codes by Goppa Pdf

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day. that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering: and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Author : Vera Pless,William Cary Huffman,Richard A. Brualdi
Publisher : North Holland
Page : 1220 pages
File Size : 44,7 Mb
Release : 1998-11-16
Category : Computers
ISBN : UOM:39015064885752

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Handbook of Coding Theory by Vera Pless,William Cary Huffman,Richard A. Brualdi Pdf

Author : Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin
Publisher : American Mathematical Society
Page : 338 pages
File Size : 54,5 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 : Carlos Moreno
Publisher : Cambridge University Press
Page : 264 pages
File Size : 54,7 Mb
Release : 1993-10-14
Category : Mathematics
ISBN : 052145901X

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Algebraic Curves Over Finite Fields by Carlos Moreno Pdf

Develops the theory of algebraic curves over finite fields, their zeta and L-functions and the theory of algebraic geometric Goppa codes.

Author : J. van Lint,G. van der Geer
Publisher : Birkhäuser
Page : 82 pages
File Size : 51,9 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783034892865

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Introduction to Coding Theory and Algebraic Geometry by J. van Lint,G. van der Geer Pdf

These notes are based on lectures given in the semmar on "Coding Theory and Algebraic Geometry" held at Schloss Mickeln, Diisseldorf, November 16-21, 1987. In 1982 Tsfasman, Vladut and Zink, using algebraic geometry and ideas of Goppa, constructed a seqeunce of codes that exceed the Gilbert-Varshamov bound. The result was considered sensational. Furthermore, it was surprising to see these unrelated areas of mathematics collaborating. The aim of this course is to give an introduction to coding theory and to sketch the ideas of algebraic geometry that led to the new result. Finally, a number of applications of these methods of algebraic geometry to coding theory are given. Since this is a new area, there are presently no references where one can find a more extensive treatment of all the material. However, both for algebraic geometry and for coding theory excellent textbooks are available. The combination ofthe two subjects can only be found in a number ofsurvey papers. A book by C. Moreno with a complete treatment of this area is in preparation. We hope that these notes will stimulate further research and collaboration of algebraic geometers and coding theorists. G. van der Geer, J.H. van Lint Introduction to CodingTheory and Algebraic Geometry PartI -- CodingTheory Jacobus H. vanLint 11 1. Finite fields In this chapter we collect (without proof) the facts from the theory of finite fields that we shall need in this course

Author : San Ling,Huaxiong Wang,Chaoping Xing
Publisher : CRC Press
Page : 340 pages
File Size : 49,9 Mb
Release : 2013-06-13
Category : Computers
ISBN : 9781420079470

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Algebraic Curves in Cryptography by San Ling,Huaxiong Wang,Chaoping Xing Pdf

The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sh

Author : Henning Stichtenoth,Michael A. Tsfasman
Publisher : Springer
Page : 235 pages
File Size : 46,8 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540472674

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Coding Theory and Algebraic Geometry by Henning Stichtenoth,Michael A. Tsfasman Pdf

About ten years ago, V.D. Goppa found a surprising connection between the theory of algebraic curves over a finite field and error-correcting codes. The aim of the meeting "Algebraic Geometry and Coding Theory" was to give a survey on the present state of research in this field and related topics. The proceedings contain research papers on several aspects of the theory, among them: Codes constructed from special curves and from higher-dimensional varieties, Decoding of algebraic geometric codes, Trace codes, Exponen- tial sums, Fast multiplication in finite fields, Asymptotic number of points on algebraic curves, Sphere packings.