Algebraic Geometric Codes

Author by : M. Tsfasman
Languange : en
Publisher by : Springer Science & Business Media
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Description : '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.


Algebraic Geometry Codes Advanced Chapters

Author by : Michael Tsfasman
Languange : en
Publisher by : American Mathematical Soc.
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Description : 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.


Advances In Algebraic Geometry Codes

Author by : Edgar Martinez-Moro
Languange : en
Publisher by : World Scientific
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Description : 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.


Algebraic Geometric Codes

Author by : Daisy Anne Amelia Wise
Languange : en
Publisher by : Unknown
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Algebraic Geometric Codes Over Rings

Author by : Judy Leavitt Walker
Languange : en
Publisher by : Unknown
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Description : The techniques of algebraic geometry have been widely and successfully applied to the study of linear codes over finite fields since the early 1980's. Recently, there has been an increased interest in the study of linear codes over finite rings. In this thesis, we combine these two approaches to coding theory by introducing and studying algebraic geometric codes over rings. We define algebraic geometric codes over any local Artinian ring A and compute their parameters. Under the additional hypothesis that A is a Gorenstein ring, we show that the class of codes we have defined is closed under duals. We show that the coordinatewise projection of an algebraic geometric code defined over A is an algebraic geometric code defined over the residue field of A. As an example of our construction, we show that the linear $doubz$/4-code which Hammons, et al., project nonlinearly to obtain the Nordstrom-Robinson code is an algebraic geometric code. In the case where A is either $doubz$/q or a Galois ring, we find an expression for the minimum Euclidean weights of (trace codes of) certain algebraic geometric codes over A in terms of an exponential sum.


Forward Error Correction Based On Algebraic Geometric Theory

Author by : Jafar A. Alzubi
Languange : en
Publisher by : Springer
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Description : This book covers the design, construction, and implementation of algebraic-geometric codes from Hermitian curves. Matlab simulations of algebraic-geometric codes and Reed-Solomon codes compare their bit error rate using different modulation schemes over additive white Gaussian noise channel model. Simulation results of Algebraic-geometric codes bit error rate performance using quadrature amplitude modulation (16QAM and 64QAM) are presented for the first time and shown to outperform Reed-Solomon codes at various code rates and channel models. The book proposes algebraic-geometric block turbo codes. It also presents simulation results that show an improved bit error rate performance at the cost of high system complexity due to using algebraic-geometric codes and Chase-Pyndiah’s algorithm simultaneously. The book proposes algebraic-geometric irregular block turbo codes (AG-IBTC) to reduce system complexity. Simulation results for AG-IBTCs are presented for the first time.


Introduction To Coding Theory And Algebraic Geometry

Author by : J. van Lint
Languange : en
Publisher by : Birkhäuser
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Description : 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.


Algebraic Geometric Codes

Author by : Michael A. Tsfasman
Languange : en
Publisher by : American Mathematical Soc.
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Codes On Algebraic Curves

Author by : Serguei A. Stepanov
Languange : en
Publisher by : Springer Science & Business Media
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Description : 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.


Linear And Algebraic Geometric Codes

Author by : Melanie Reed Smith
Languange : en
Publisher by : Unknown
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Description : "This paper gives a brief introduction to linear and algebraic-geometric codes. Some basic results are covered from finite field theory before going into coding theory. Linear codes are discussed, and four well-known codes are presented. Next is background material from algebraic geometry. Finally, algebraic-geometric codes are introduced and their improvement over other codes discussed"--Document.


Coding Theory And Algebraic Geometry

Author by : Henning Stichtenoth
Languange : en
Publisher by : Springer
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Description : 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.


Algebraic Geometry For Coding Theory And Cryptography

Author by : Everett W. Howe
Languange : en
Publisher by : Springer
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Description : Covering topics in algebraic geometry, coding theory, and cryptography, this volume presents interdisciplinary group research completed for the February 2016 conference at the Institute for Pure and Applied Mathematics (IPAM) in cooperation with the Association for Women in Mathematics (AWM). The conference gathered research communities across disciplines to share ideas and problems in their fields and formed small research groups made up of graduate students, postdoctoral researchers, junior faculty, and group leaders who designed and led the projects. Peer reviewed and revised, each of this volume's five papers achieves the conference’s goal of using algebraic geometry to address a problem in either coding theory or cryptography. Proposed variants of the McEliece cryptosystem based on different constructions of codes, constructions of locally recoverable codes from algebraic curves and surfaces, and algebraic approaches to the multicast network coding problem are only some of the topics covered in this volume. Researchers and graduate-level students interested in the interactions between algebraic geometry and both coding theory and cryptography will find this volume valuable.


Decoding Algebraic Geometric Codes

Author by : José Ignacio Farrán
Languange : en
Publisher by : Unknown
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Algebraic Geometry Codes

Author by : Michael A. Tsfasman
Languange : en
Publisher by : Unknown
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Description : This 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, and dense packings.


Algebraic Geometry Modeling In Information Theory

Author by : Edgar Martinez-Moro
Languange : en
Publisher by : World Scientific
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Total Read : 28
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Description : Algebraic & geometry methods have constituted a basic background and tool for people working on classic block coding theory and cryptography. Nowadays, new paradigms on coding theory and cryptography have arisen such as: Network coding, S-Boxes, APN Functions, Steganography and decoding by linear programming. Again understanding the underlying procedure and symmetry of these topics needs a whole bunch of non trivial knowledge of algebra and geometry that will be used to both, evaluate those methods and search for new codes and cryptographic applications. This book shows those methods in a self-contained form.


Gr Bner Bases Coding And Cryptography

Author by : Massimiliano Sala
Languange : en
Publisher by : Springer Science & Business Media
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Description : Coding theory and cryptography allow secure and reliable data transmission, which is at the heart of modern communication. Nowadays, it is hard to find an electronic device without some code inside. Gröbner bases have emerged as the main tool in computational algebra, permitting numerous applications, both in theoretical contexts and in practical situations. This book is the first book ever giving a comprehensive overview on the application of commutative algebra to coding theory and cryptography. For example, all important properties of algebraic/geometric coding systems (including encoding, construction, decoding, list decoding) are individually analysed, reporting all significant approaches appeared in the literature. Also, stream ciphers, PK cryptography, symmetric cryptography and Polly Cracker systems deserve each a separate chapter, where all the relevant literature is reported and compared. While many short notes hint at new exciting directions, the reader will find that all chapters fit nicely within a unified notation.


Coding Theory And Number Theory

Author by : T. Hiramatsu
Languange : en
Publisher by : Springer Science & Business Media
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Description : This book grew out of our lectures given in the Oberseminar on 'Cod ing Theory and Number Theory' at the Mathematics Institute of the Wiirzburg University in the Summer Semester, 2001. The coding the ory combines mathematical elegance and some engineering problems to an unusual degree. The major advantage of studying coding theory is the beauty of this particular combination of mathematics and engineering. In this book we wish to introduce some practical problems to the math ematician and to address these as an essential part of the development of modern number theory. The book consists of five chapters and an appendix. Chapter 1 may mostly be dropped from an introductory course of linear codes. In Chap ter 2 we discuss some relations between the number of solutions of a diagonal equation over finite fields and the weight distribution of cyclic codes. Chapter 3 begins by reviewing some basic facts from elliptic curves over finite fields and modular forms, and shows that the weight distribution of the Melas codes is represented by means of the trace of the Hecke operators acting on the space of cusp forms. Chapter 4 is a systematic study of the algebraic-geometric codes. For a long time, the study of algebraic curves over finite fields was the province of pure mathematicians. In the period 1977 - 1982, V. D. Goppa discovered an amazing connection between the theory of algebraic curves over fi nite fields and the theory of q-ary codes.


Algebraic Geometric Codes From Elliptic Curves

Author by : David Jaramillo Martínez
Languange : en
Publisher by : Unknown
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Description : "Let C=[n,k,d] be a Goppa Code constructed from an elliptic curve. It is known that C is an AMDS (almost MDS) code i.e. d=n-k. By studying how many information sets C has (an MDS Code has \binom{n}{k} information sets) we investigate, for a given rate \frac{k}{n}, how close are actually Goppa Codes from being MDS, having in mind the benefit that they do not require such a big underlying field as say Reed-Solomon Codes. For the case k=3 we say exactly how far are them of being MDS."--Tomado del Formato de Documento de Grado.


Coding Theory And Algebraic Geometry

Author by : Henning Stichtenoth
Languange : en
Publisher by : Springer
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Total Read : 30
Total Download : 156
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Description : 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.


Algebraic Function Fields And Codes

Author by : Henning Stichtenoth
Languange : en
Publisher by : Springer Science & Business Media
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Total Read : 55
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Description : 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.


Codes And Curves

Author by : Judy L. Walker
Languange : en
Publisher by : American Mathematical Soc.
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Description : When information is transmitted, errors are likely to occur. Coding theory examines efficient ways of packaging data so that these errors can be detected, or even corrected. The traditional tools of coding theory have come from combinatorics and group theory. Lately, however, coding theorists have added techniques from algebraic geometry to their toolboxes. In particular, by re-interpreting the Reed-Solomon codes, one can see how to define new codes based on divisors on algebraic curves. For instance, using modular curves over finite fields, Tsfasman, Vladut, and Zink showed that one can define a sequence of codes with asymptotically better parameters than any previously known codes. This monograph is based on a series of lectures the author gave as part of the IAS/PCMI program on arithmetic algebraic geometry. Here, the reader is introduced to the exciting field of algebraic geometric coding theory. Presenting the material in the same conversational tone of the lectures, the author covers linear codes, including cyclic codes, and both bounds and asymptotic bounds on the parameters of codes. 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 is discussed. This book is published in cooperation with IAS/Park City Mathematics Institute.


Algebraic Geometric Codes On Anticanonical Surfaces

Author by : Jennifer A. Davis
Languange : en
Publisher by : Unknown
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Description : Algebraic geometric codes (or AG codes) provide a way to correct errors that occur during the transmission of digital information. AG codes on curves have been studied extensively, but much less work has been done for AG codes on higher dimensional varieties. In particular, we seek good bounds for the minimum distance.


Codes Curves And Signals

Author by : Alexander Vardy
Languange : en
Publisher by : Springer Science & Business Media
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Description : Codes, Curves, and Signals: Common Threads in Communications is a collection of seventeen contributions from leading researchers in communications. The book provides a representative cross-section of cutting edge contemporary research in the fields of algebraic curves and the associated decoding algorithms, the use of signal processing techniques in coding theory, and the application of information-theoretic methods in communications and signal processing. The book is organized into three parts: Curves and Codes, Codes and Signals, and Signals and Information. Codes, Curves, and Signals: Common Threads in Communications is a tribute to the broad and profound influence of Richard E. Blahut on the fields of algebraic coding, information theory, and digital signal processing. All the contributors have individually and collectively dedicated their work to R. E. Blahut. Codes, Curves, and Signals: Common Threads in Communications is an excellent reference for researchers and professionals.