Codes On Algebraic Curves

Author by : Serguei A. Stepanov
Languange : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 61
Total Download : 422
File Size : 51,5 Mb
GET BOOK

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.


Codes And Algebraic Curves

Author by : Oliver Pretzel
Languange : en
Publisher by : Clarendon Press
Format Available : PDF, ePub, Mobi
Total Read : 49
Total Download : 842
File Size : 55,9 Mb
GET BOOK

Description : 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.


Codes And Curves

Author by : Judy L. Walker
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 83
Total Download : 990
File Size : 49,6 Mb
GET BOOK

Description : This book is based on a series of lectures the author gave as part of the IAS/Park City Mathematics Institute (Utah) program on arithmetic algebraic geometry. It introduces the reader 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. This book is published in cooperation with IAS/Park City Mathematics Institute.


Group Codes On Algebraic Curves

Author by : J. P. Hansen
Languange : en
Publisher by : Unknown
Format Available : PDF, ePub, Mobi
Total Read : 80
Total Download : 765
File Size : 54,8 Mb
GET BOOK

Description :


Algebraic Geometry Codes Advanced Chapters

Author by : Michael Tsfasman
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 27
Total Download : 164
File Size : 41,8 Mb
GET BOOK

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.


Group Codes On Algebraic Curves

Author by : Johan P. Hansen
Languange : de
Publisher by : Unknown
Format Available : PDF, ePub, Mobi
Total Read : 31
Total Download : 145
File Size : 51,7 Mb
GET BOOK

Description :


Algebraic Geometric Codes

Author by : M. Tsfasman
Languange : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 14
Total Download : 958
File Size : 55,6 Mb
GET BOOK

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 Curves And Their Applications

Author by : Lubjana Beshaj
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 58
Total Download : 705
File Size : 46,6 Mb
GET BOOK

Description : This volume contains a collection of papers on algebraic curves and their applications. While algebraic curves traditionally have provided a path toward modern algebraic geometry, they also provide many applications in number theory, computer security and cryptography, coding theory, differential equations, and more. Papers cover topics such as the rational torsion points of elliptic curves, arithmetic statistics in the moduli space of curves, combinatorial descriptions of semistable hyperelliptic curves over local fields, heights on weighted projective spaces, automorphism groups of curves, hyperelliptic curves, dessins d'enfants, applications to Painlevé equations, descent on real algebraic varieties, quadratic residue codes based on hyperelliptic curves, and Abelian varieties and cryptography. This book will be a valuable resource for people interested in algebraic curves and their connections to other branches of mathematics.


Advances In Algebraic Geometry Codes

Author by : Edgar Martinez-Moro
Languange : en
Publisher by : World Scientific
Format Available : PDF, ePub, Mobi
Total Read : 16
Total Download : 802
File Size : 48,9 Mb
GET BOOK

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 Curves In Cryptography

Author by : San Ling
Languange : en
Publisher by : CRC Press
Format Available : PDF, ePub, Mobi
Total Read : 28
Total Download : 212
File Size : 46,5 Mb
GET BOOK

Description : 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


Algebraic Geometry For Coding Theory And Cryptography

Author by : Everett W. Howe
Languange : en
Publisher by : Springer
Format Available : PDF, ePub, Mobi
Total Read : 68
Total Download : 511
File Size : 41,8 Mb
GET BOOK

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.


Mat Report

Author by : Anonim
Languange : en
Publisher by : Unknown
Format Available : PDF, ePub, Mobi
Total Read : 17
Total Download : 785
File Size : 42,6 Mb
GET BOOK

Description :


Algebraic Curves Over Finite Fields

Author by : Carlos Moreno
Languange : en
Publisher by : Cambridge University Press
Format Available : PDF, ePub, Mobi
Total Read : 34
Total Download : 972
File Size : 51,9 Mb
GET BOOK

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


Groip Codes On Algebraic Curves

Author by : Johan P. Hansen
Languange : en
Publisher by : Unknown
Format Available : PDF, ePub, Mobi
Total Read : 86
Total Download : 422
File Size : 41,6 Mb
GET BOOK

Description :


Coding Theory And Algebraic Geometry

Author by : Henning Stichtenoth
Languange : en
Publisher by : Springer
Format Available : PDF, ePub, Mobi
Total Read : 73
Total Download : 243
File Size : 55,8 Mb
GET BOOK

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 Curves Over A Finite Field

Author by : J. W. P. Hirschfeld
Languange : en
Publisher by : Princeton University Press
Format Available : PDF, ePub, Mobi
Total Read : 17
Total Download : 405
File Size : 42,9 Mb
GET BOOK

Description : This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.


Implementation Of A Decoding Algorithm For Codes From Algebraic Curves In The Programming Language Sage

Author by : Anonim
Languange : en
Publisher by : Unknown
Format Available : PDF, ePub, Mobi
Total Read : 18
Total Download : 161
File Size : 50,8 Mb
GET BOOK

Description : Creating and implementing efficient decoding algorithms is an important study in Coding Theory. This thesis focuses on the decoding algorithms for Reed-Solomon Codes and Hermitian Codes, specifically the Berlekamp-Massey Algorithm and the Berlekamp-Massey- Sakata Algorithm. The Berlekamp-Massey-Sakata Algorithm alone is not enough to decode up to the minimum distance bound so Feng and Rao's technique of Majority Voting is included to allow decoding up to the minimum distance bound and even beyond for some high rate codes. An implementation for each of these decoding algorithms using the programming language Sage was created with the goal that these could be made available to the community of Sage users.


Coding Theory And Number Theory

Author by : T. Hiramatsu
Languange : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 94
Total Download : 112
File Size : 41,6 Mb
GET BOOK

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.


Forward Error Correction Based On Algebraic Geometric Theory

Author by : Jafar A. Alzubi
Languange : en
Publisher by : Springer
Format Available : PDF, ePub, Mobi
Total Read : 89
Total Download : 517
File Size : 43,8 Mb
GET BOOK

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.


Plane Algebraic Curves

Author by : Gerd Fischer
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 93
Total Download : 273
File Size : 47,7 Mb
GET BOOK

Description : This is an excellent introduction to algebraic geometry, which assumes only standard undergraduate mathematical topics: complex analysis, rings and fields, and topology. Reading this book will help establish the geometric intuition that lies behind the more advanced ideas and techniques used in the study of higher-dimensional varieties.