Error correcting Codes and Finite Fields

Starting with the elementary ideas of parity check codes, this work takes the reader via BCH and Reed-Solomon codes all the way to the geometric Goppa codes. The necessary mathematics is developed in parallel with the applications.

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Author: Oliver Pretzel

Publisher: Oxford University Press

ISBN: STANFORD:36105000110978

Category: Computers

Page: 398

View: 895

Starting with the elementary ideas of parity check codes, this work takes the reader via BCH and Reed-Solomon codes all the way to the geometric Goppa codes. The necessary mathematics is developed in parallel with the applications.

Error Correcting Codes

Discusses information theory, finite fields, classical error correcting codes, codes and combinatorics, and tables and curves

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Author: Alain Poli

Publisher:

ISBN: UOM:39015025385520

Category: Computers

Page: 512

View: 848

Discusses information theory, finite fields, classical error correcting codes, codes and combinatorics, and tables and curves

A Course in Algebraic Error Correcting Codes

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.

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Author: Simeon Ball

Publisher: Springer Nature

ISBN: 9783030411534

Category: Mathematics

Page: 177

View: 546

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.

Introduction to the Theory of Error Correcting Codes

4.1 GROUPS As we saw in Chapter 3, finite fields are very useful tools for constructing multiple-error-correcting codes. A knowledge of finite fields is useful in order to understand the important class of cyclic codes.

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Author: Vera Pless

Publisher: John Wiley & Sons

ISBN: 9781118030998

Category: Mathematics

Page: 224

View: 257

A complete introduction to the many mathematical tools used tosolve practical problems in coding. Mathematicians have been fascinated with the theory oferror-correcting codes since the publication of Shannon's classicpapers fifty years ago. With the proliferation of communicationssystems, computers, and digital audio devices that employerror-correcting codes, the theory has taken on practicalimportance in the solution of coding problems. This solutionprocess requires the use of a wide variety of mathematical toolsand an understanding of how to find mathematical techniques tosolve applied problems. Introduction to the Theory of Error-Correcting Codes, Third Editiondemonstrates this process and prepares students to cope with codingproblems. Like its predecessor, which was awarded a three-starrating by the Mathematical Association of America, this updated andexpanded edition gives readers a firm grasp of the timelessfundamentals of coding as well as the latest theoretical advances.This new edition features: * A greater emphasis on nonlinear binary codes * An exciting new discussion on the relationship between codes andcombinatorial games * Updated and expanded sections on the Vashamov-Gilbert bound, vanLint-Wilson bound, BCH codes, and Reed-Muller codes * Expanded and updated problem sets. Introduction to the Theory of Error-Correcting Codes, Third Editionis the ideal textbook for senior-undergraduate and first-yeargraduate courses on error-correcting codes in mathematics, computerscience, and electrical engineering.

An Introduction to Error Correcting Codes with Applications

5. 2 Rings and Ideals 148 5. 3 Ideals and Cyclic Subspaces 152 5. 4 Generator Matrices and Parity-Check Matrices 159 5. 5 Encoding Cyclic Codest 163 5. 6 Syndromes and Simple Decoding Procedures 168 5. 7 Burst Error Correcting 175 5. 8 ...

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Author: Scott A. Vanstone

Publisher: Springer Science & Business Media

ISBN: 9781475720327

Category: Technology & Engineering

Page: 289

View: 104

5. 2 Rings and Ideals 148 5. 3 Ideals and Cyclic Subspaces 152 5. 4 Generator Matrices and Parity-Check Matrices 159 5. 5 Encoding Cyclic Codest 163 5. 6 Syndromes and Simple Decoding Procedures 168 5. 7 Burst Error Correcting 175 5. 8 Finite Fields and Factoring xn-l over GF(q) 181 5. 9 Another Method for Factoring xn-l over GF(q)t 187 5. 10 Exercises 193 Chapter 6 BCH Codes and Bounds for Cyclic Codes 6. 1 Introduction 201 6. 2 BCH Codes and the BCH Bound 205 6. 3 Bounds for Cyclic Codest 210 6. 4 Decoding BCH Codes 215 6. 5 Linearized Polynomials and Finding Roots of Polynomialst 224 6. 6 Exercises 231 Chapter 7 Error Correction Techniques and Digital Audio Recording 7. 1 Introduction 237 7. 2 Reed-Solomon Codes 237 7. 3 Channel Erasures 240 7. 4 BCH Decoding with Erasures 244 7. 5 Interleaving 250 7. 6 Error Correction and Digital Audio Recording 256 7.

Error Correcting Codes

Introducing the reader to the basics of coding theory, this text covers practical and theoretical computing and problem-solving.

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Author: John Baylis

Publisher: Chapman & Hall/CRC

ISBN: 1138416088

Category:

Page:

View: 343

Assuming little previous mathematical knowledge, Error Correcting Codes provides a sound introduction to key areas of the subject. Topics have been chosen for their importance and practical significance, which Baylis demonstrates in a rigorous but gentle mathematical style. Coverage includes optimal codes; linear and non-linear codes; general techniques of decoding errors and erasures; error detection; syndrome decoding, and much more. Error Correcting Codes contains not only straight maths, but also exercises on more investigational problem solving. Chapters on number theory and polynomial algebra are included to support linear codes and cyclic codes, and an extensive reminder of relevant topics in linear algebra is given. Exercises are placed within the main body of the text to encourage active participation by the reader, with comprehensive solutions provided. Error Correcting Codes will appeal to undergraduate students in pure and applied mathematical fields, software engineering, communications engineering, computer science and information technology, and to organizations with substantial research and development in those areas.

VLSI Architectures for Modern Error Correcting Codes

Moreover, numerous coding books, such as [4, 5, 6], have chapters introducing the basics of finite fields. The implementation of finite field arithmetic largely affects the hardware complexities of the error-correcting encoders and ...

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Author: Xinmiao Zhang

Publisher: CRC Press

ISBN: 9781482229653

Category: Technology & Engineering

Page: 410

View: 917

Error-correcting codes are ubiquitous. They are adopted in almost every modern digital communication and storage system, such as wireless communications, optical communications, Flash memories, computer hard drives, sensor networks, and deep-space probing. New-generation and emerging applications demand codes with better error-correcting capability. On the other hand, the design and implementation of those high-gain error-correcting codes pose many challenges. They usually involve complex mathematical computations, and mapping them directly to hardware often leads to very high complexity. VLSI Architectures for Modern Error-Correcting Codes serves as a bridge connecting advancements in coding theory to practical hardware implementations. Instead of focusing on circuit-level design techniques, the book highlights integrated algorithmic and architectural transformations that lead to great improvements on throughput, silicon area requirement, and/or power consumption in the hardware implementation. The goal of this book is to provide a comprehensive and systematic review of available techniques and architectures, so that they can be easily followed by system and hardware designers to develop en/decoder implementations that meet error-correcting performance and cost requirements. This book can be also used as a reference for graduate-level courses on VLSI design and error-correcting coding. Particular emphases are placed on hard- and soft-decision Reed-Solomon (RS) and Bose-Chaudhuri-Hocquenghem (BCH) codes, and binary and non-binary low-density parity-check (LDPC) codes. These codes are among the best candidates for modern and emerging applications due to their good error-correcting performance and lower implementation complexity compared to other codes. To help explain the computations and en/decoder architectures, many examples and case studies are included. More importantly, discussions are provided on the advantages and drawbacks of different implementation approaches and architectures.

Fault Tolerant Signal Processing Using Finite Fields and Error correcting Codes

Finite field arithmetic may be efficiently applied for implementing signal processing architectures. The fundamental theoretical work in this area was done by Matluk and Gill.

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Author: G. R. Redinbo

Publisher:

ISBN: OCLC:227596749

Category:

Page: 98

View: 724

Finite field arithmetic may be efficiently applied for implementing signal processing architectures. The fundamental theoretical work in this area was done by Matluk and Gill. This report presents several new approaches for incorporating error-correcting codes with signal processing operations yielding fault tolerant systems. Fault tolerant levels can be distributed throughout the system's architecture. Such architectures will become necessary when very sophisticated and dense systems are implemented with Very Large Scale Integration.

Applied Algebra Algebraic Algorithms and Error Correcting Codes

We introduce a new class of quantum errorcorrecting codes derived from (classical) Reed–Solomon codes over finite fields of characteristic two. Quantum circuits for encoding and decoding based on the discrete cyclic Fourier transform ...

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Author: Marc Fossorier

Publisher: Springer

ISBN: 9783540467960

Category: Computers

Page: 518

View: 499

This book constitutes the refereed proceedings of the 19th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-13, held in Honolulu, Hawaii, USA in November 1999. The 42 revised full papers presented together with six invited survey papers were carefully reviewed and selected from a total of 86 submissions. The papers are organized in sections on codes and iterative decoding, arithmetic, graphs and matrices, block codes, rings and fields, decoding methods, code construction, algebraic curves, cryptography, codes and decoding, convolutional codes, designs, decoding of block codes, modulation and codes, Gröbner bases and AG codes, and polynomials.

Introduction to the Theory of Error correcting Codes

An introduction to the theory of error-correction codes, and in particular to linear block codes is provided in this book.

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Author: Vera Pless

Publisher: Wiley-Interscience

ISBN: UOM:39015015706206

Category: Error-correcting codes (Information theory).

Page: 201

View: 648

An introduction to the theory of error-correction codes, and in particular to linear block codes is provided in this book. It considers such codes as Hamming codes and Golay codes, correction of double errors, use of finite fields, cyclic codes, BCH codes and weight distributions, as well as design of codes. In this second edition, the author includes more material on non-binary code and cyclic codes. In addition some proofs have been simplified and there are many more examples and problems. The text has been aimed at mathematicians, electrical engineers and computer scientists.

Block Error Correcting Codes

2 Finite Fields During the last two decades more and more abstract algebraic tools such as the theory of finite fields and the theory of polynomials over finite fields have influenced coding. R. Lidl and H. Niederreiter, [10], p. 305.

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Author: Sebastian Xambo-Descamps

Publisher: Springer Science & Business Media

ISBN: 9783642189975

Category: Computers

Page: 266

View: 711

Error-correcting codes have been incorporated in numerous working communication and memory systems. This book covers the mathematical aspects of the theory of block error-correcting codes together, in mutual reinforcement, with computational discussions, implementations and examples of all relevant concepts, functions and algorithms. This combined approach facilitates the reading and understanding of the subject. The digital companion of the book is a non-printable .pdf document with hyperlinks. The examples included in the book can be run with just a mouse click and modified and saved by users for their own purpose.

Algebraic Coding Theory Revised Edition

This is the revised edition of Berlekamp's famous book, 'Algebraic Coding Theory', originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field.

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Author: Elwyn R Berlekamp

Publisher: World Scientific

ISBN: 9789814635912

Category: Mathematics

Page: 500

View: 263

This is the revised edition of Berlekamp's famous book, 'Algebraic Coding Theory', originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field. One of these is an algorithm for decoding Reed-Solomon and Bose-Chaudhuri-Hocquenghem codes that subsequently became known as the Berlekamp-Massey Algorithm. Another is the Berlekamp algorithm for factoring polynomials over finite fields, whose later extensions and embellishments became widely used in symbolic manipulation systems. Other novel algorithms improved the basic methods for doing various arithmetic operations in finite fields of characteristic two. Other major research contributions in this book included a new class of Lee metric codes, and precise asymptotic results on the number of information symbols in long binary BCH codes.Selected chapters of the book became a standard graduate textbook.Both practicing engineers and scholars will find this book to be of great value.

Error correcting Codes Finite Geometries and Cryptography

Conference on Error-control Codes, Information Theory, and Applied Cryptography, December 5-6, 2007, Fields Institute, Toronto, Ontario, Canada : Canadian Mathematical Society Special Session on Error Control Codes, Information Theory, ...

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Author: Aiden A. Bruen

Publisher: American Mathematical Soc.

ISBN: 9780821849569

Category: Mathematics

Page: 244

View: 110

This interdisciplinary volume contains papers from both a conference and special session on Error-Control Codes, Information Theory and Applied Cryptography. The conference was held at the Fields Institute in Toronto, On, Canada from December 5-6, 2007, and the special session was held at the Canadian Mathematical Society's winter meeting in London, ON, Canada from December 8-10, 2007. The volume features cutting-edge theoretical results on the Reed-Muller and Reed-Solomon codes, classical linear codes, codes from nets and block designs, LDPC codes, perfect quantum and orthogonal codes, iterative decoding, magnetic storage and digital memory devices, and MIMO channels. There are new contributions on privacy reconciliation, resilient functions, cryptographic hash functions, and new work on quantum coins. Related original work in finite geometries concerns two-weight codes coming from partial spreads, (0, 1) matrices with forbidden configurations, Andre embeddings, and representations of projective spaces in affine planes. Great care has been taken to ensure that high expository standards are met by the papers in this volume. Accordingly, the papers are written in a user-friendly format. The hope is that this volume will be of interst and of benefit both to the experienced and to newcomers alike.

The Theory of Error Correcting Codes

To find a square root: l0)l/2 : (a5)l/2 : (a20)l/2 = am = We shall see in Chapter 4 that any finite field can be constructed in exactly the same way, and has the property that the multiplicative group of nonzero elements is cyclic, ...

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Author: Florence Jessie MacWilliams

Publisher: Elsevier

ISBN: 9780444850102

Category: Electronic books

Page: 762

View: 859

Error Correction Coding

6.3 Decoding BCH and RS Codes: The General Outline 255 we'll set the problem up first including the error values. ... polynomials over finite fields (see, e.g., [33, 472]), but for the fields usually used for error-correction codes and ...

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Author: Todd K. Moon

Publisher: John Wiley & Sons

ISBN: 9781119567479

Category: Computers

Page: 992

View: 858

Providing in-depth treatment of error correction Error Correction Coding: Mathematical Methods and Algorithms, 2nd Edition provides a comprehensive introduction to classical and modern methods of error correction. The presentation provides a clear, practical introduction to using a lab-oriented approach. Readers are encouraged to implement the encoding and decoding algorithms with explicit algorithm statements and the mathematics used in error correction, balanced with an algorithmic development on how to actually do the encoding and decoding. Both block and stream (convolutional) codes are discussed, and the mathematics required to understand them are introduced on a “just-in-time” basis as the reader progresses through the book. The second edition increases the impact and reach of the book, updating it to discuss recent important technological advances. New material includes: Extensive coverage of LDPC codes, including a variety of decoding algorithms. A comprehensive introduction to polar codes, including systematic encoding/decoding and list decoding. An introduction to fountain codes. Modern applications to systems such as HDTV, DVBT2, and cell phones Error Correction Coding includes extensive program files (for example, C++ code for all LDPC decoders and polar code decoders), laboratory materials for students to implement algorithms, and an updated solutions manual, all of which are perfect to help the reader understand and retain the content. The book covers classical BCH, Reed Solomon, Golay, Reed Muller, Hamming, and convolutional codes which are still component codes in virtually every modern communication system. There are also fulsome discussions of recently developed polar codes and fountain codes that serve to educate the reader on the newest developments in error correction.

Foundations of Coding

The definition-theorem proof style used in mathematics texts is employed through the book but formalism is avoided wherever possible.

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Author: Jiri Adamek

Publisher: John Wiley & Sons

ISBN: 9781118031513

Category: Computers

Page: 352

View: 395

Although devoted to constructions of good codes for error control, secrecy or data compression, the emphasis is on the first direction. Introduces a number of important classes of error-detecting and error-correcting codes as well as their decoding methods. Background material on modern algebra is presented where required. The role of error-correcting codes in modern cryptography is treated as are data compression and other topics related to information theory. The definition-theorem proof style used in mathematics texts is employed through the book but formalism is avoided wherever possible.

Application of Fourier Transform Over Finite Fields to Error correcting Codes

BCH code is one of the most powerful and the most extensively studied classes of algebraic error-correcting codes.

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Author: David Mun-Hien Choy

Publisher:

ISBN: OCLC:227583340

Category: Coding theory

Page: 184

View: 638

BCH code is one of the most powerful and the most extensively studied classes of algebraic error-correcting codes. Unfortunately, it was found that its performance in error-correction deteriorates as its length increases. In this report, a class of algebraic linear codes is proposed, based on the structure of Fourier Transform over finite fields. This class of codes includes the BCH codes, and Srivastava codes as proper subclasses. Several constructive bounds on the minimum distance of these codes are derived and are shown to be achievable using Berlekamp's iterative decoding algorithm, or using Goppa's method based on divided difference. A new distance bound is also obtained for a group of binary Srivastava codes. (Modified author abstract).

Finite Fields with Applications to Coding Theory Cryptography and Related Areas

Proceedings of the Sixth International Conference on Finite Fields and Applications, held at Oaxaca, México, May 21–25, 2001 Gary L. Mullen, Henning Stichtenoth, Horacio Tapia-Recillas. New Quantum Error-Correcting Codes from Hermitian ...

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Author: Gary L. Mullen

Publisher: Springer Science & Business Media

ISBN: 9783642594359

Category: Mathematics

Page: 335

View: 177

The Sixth International Conference on Finite Fields and Applications, Fq6, held in the city of Oaxaca, Mexico, from May 21-25, 2001, continued a series of biennial international conferences on finite fields. This volume documents the steadily increasing interest in this topic. Finite fields are an important tool in discrete mathematics and its applications cover algebraic geometry, coding theory, cryptology, design theory, finite geometries, and scientific computation, among others. An important feature is the interplay between theory and applications which has led to many new perspectives in research on finite fields and other areas. This interplay has been emphasized in this series of conferences and certainly was reflected in Fq6. This volume offers up-to-date original research papers by leading experts in the area.

Applied Algebra Algebraic Algorithms and Error Correcting Codes

This result is based on the existence of certain number fields that have an infinite class field tower in which some primes of small norm split completely. 1 Introduction Algebraic Error-correcting Codes. For a finite field Fq, an [n, ...

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Author: Serdar Boztas

Publisher: Springer

ISBN: 9783540456247

Category: Mathematics

Page: 404

View: 379

The AAECC Symposia Series was started in 1983 by Alain Poli (Toulouse), who, together with R. Desq, D. Lazard, and P. Camion, organized the ?rst conference. Originally the acronym AAECC meant “Applied Algebra and Error-Correcting Codes”. Over the years its meaning has shifted to “Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes”, re?ecting the growing importance of complexity in both decoding algorithms and computational algebra. AAECC aims to encourage cross-fertilization between algebraic methods and their applications in computing and communications. The algebraic orientation is towards ?nite ?elds, complexity, polynomials, and graphs. The applications orientation is towards both theoretical and practical error-correction coding, and, since AAECC 13 (Hawaii, 1999), towards cryptography. AAECC was the ?rst symposium with papers connecting Gr ̈obner bases with E-C codes. The balance between theoretical and practical is intended to shift regularly; at AAECC-14 the focus was on the theoretical side. The main subjects covered were: – Codes: iterative decoding, decoding methods, block codes, code construction. – Codes and algebra: algebraic curves, Gr ̈obner bases, and AG codes. – Algebra: rings and ?elds, polynomials. – Codes and combinatorics: graphs and matrices, designs, arithmetic. – Cryptography. – Computational algebra: algebraic algorithms. – Sequences for communications.