Error Correcting Linear Codes

Error Correcting Linear Codes

This text offers an introduction to error-correcting linear codes for researchers and graduate students in mathematics, computer science and engineering.

Author: Anton Betten

Publisher: Springer Science & Business Media

ISBN: 9783540317036

Category: Mathematics

Page: 798

View: 681

This text offers an introduction to error-correcting linear codes for researchers and graduate students in mathematics, computer science and engineering. The book differs from other standard texts in its emphasis on the classification of codes by means of isometry classes. The relevant algebraic are developed rigorously. Cyclic codes are discussed in great detail. In the last four chapters these isometry classes are enumerated, and representatives are constructed algorithmically.
Categories: Mathematics

Error correcting Codes

Error correcting Codes

The coding problem; Introduction to algebra; Linear codes; Error correction capabilities of linear codes; Important linear block codes; Polynomial rings and galois fields; Linear switching circuits; Cyclic codes; Bose-chaudhuri-hocquenghem ...

Author: William Wesley Peterson

Publisher: MIT Press

ISBN: 0262160390

Category: Computers

Page: 560

View: 216

The coding problem; Introduction to algebra; Linear codes; Error correction capabilities of linear codes; Important linear block codes; Polynomial rings and galois fields; Linear switching circuits; Cyclic codes; Bose-chaudhuri-hocquenghem codes; Arithmetic codes.
Categories: Computers

A Course in Algebraic Error Correcting Codes

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.

Author: Simeon Ball

Publisher: Springer Nature

ISBN: 9783030411534

Category: Mathematics

Page: 177

View: 554

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.
Categories: Mathematics

Fundamentals of Error Correcting Codes

Fundamentals of Error Correcting Codes

Fundamentals of Error Correcting Codes is an in-depth introduction to coding theory from both an engineering and mathematical viewpoint.

Author: W. Cary Huffman

Publisher: Cambridge University Press

ISBN: 1139439502

Category: Technology & Engineering

Page:

View: 828

Fundamentals of Error Correcting Codes is an in-depth introduction to coding theory from both an engineering and mathematical viewpoint. As well as covering classical topics, there is much coverage of techniques which could only be found in specialist journals and book publications. Numerous exercises and examples and an accessible writing style make this a lucid and effective introduction to coding theory for advanced undergraduate and graduate students, researchers and engineers, whether approaching the subject from a mathematical, engineering or computer science background.
Categories: Technology & Engineering

An Introduction to Error Correcting Codes with Applications

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 ...

Author: Scott A. Vanstone

Publisher: Springer Science & Business Media

ISBN: 9781475720327

Category: Technology & Engineering

Page: 289

View: 258

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.
Categories: Technology & Engineering

Error Correcting Codes

Error Correcting Codes

But from the proof of Theorem 6.1 this n is precisely the maximal number of
columns of length n - / with no pair of columns dependent. Hence, by definition С
is a Hamming code. Moving on to 2-error correcting linear codes, the condition
for ...

Author: D J. Baylis

Publisher: CRC Press

ISBN: 0412786907

Category: Mathematics

Page: 232

View: 114

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.
Categories: Mathematics

Introduction to the Theory of Error correcting Codes

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.

Author: Vera Pless

Publisher: Wiley-Interscience

ISBN: UOM:39015015706206

Category: Error-correcting codes (Information theory)

Page: 201

View: 792

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.
Categories: Error-correcting codes (Information theory)

An Introduction to Error correcting Codes

An Introduction to Error correcting Codes

Codes, Kodierung (Telegrafie) ; Kodierung, Datendarstellung, Bit, Byte (EDV).

Author: Shu Lin

Publisher: Prentice Hall

ISBN: UOM:39076006526870

Category: Codes correcteurs d'erreurs (Théorie de l'information)

Page: 330

View: 454

Codes, Kodierung (Telegrafie) ; Kodierung, Datendarstellung, Bit, Byte (EDV).
Categories: Codes correcteurs d'erreurs (Théorie de l'information)

Coding Theory

Coding Theory

Modern introduction to theory of coding and decoding with many exercises and examples.

Author: San Ling

Publisher: Cambridge University Press

ISBN: 0521529239

Category: Mathematics

Page: 222

View: 892

Coding theory is concerned with successfully transmitting data through a noisy channel and correcting errors in corrupted messages. It is of central importance for many applications in computer science or engineering. This book gives a comprehensive introduction to coding theory whilst only assuming basic linear algebra. It contains a detailed and rigorous introduction to the theory of block codes and moves on to more advanced topics like BCH codes, Goppa codes and Sudan's algorithm for list decoding. The issues of bounds and decoding, essential to the design of good codes, features prominently. The authors of this book have, for several years, successfully taught a course on coding theory to students at the National University of Singapore. This book is based on their experiences and provides a thoroughly modern introduction to the subject. There are numerous examples and exercises, some of which introduce students to novel or more advanced material.
Categories: Mathematics

Introduction to Coding Theory

Introduction to Coding Theory

This 2006 book introduces the theoretical foundations of error-correcting codes for senior-undergraduate to graduate students.

Author: Ron Roth

Publisher: Cambridge University Press

ISBN: 0521845041

Category: Computers

Page: 566

View: 496

This 2006 book introduces the theoretical foundations of error-correcting codes for senior-undergraduate to graduate students.
Categories: Computers

A Commonsense Approach to the Theory of Error Correcting Codes

A Commonsense Approach to the Theory of Error Correcting Codes

This text explains the basic circuits in a refreshingly practical way thatwill appeal to undergraduate electrical engineering students as well as to engineers and techniciansworking in industry.Arazi's truly commonsense approach provides a ...

Author: Benjamin Arazi

Publisher: MIT Press

ISBN: 0262010984

Category: Computers

Page: 208

View: 236

Teaching the theory of error correcting codes on an introductory level is a difficulttask. The theory, which has immediate hardware applications, also concerns highly abstractmathematical concepts. This text explains the basic circuits in a refreshingly practical way thatwill appeal to undergraduate electrical engineering students as well as to engineers and techniciansworking in industry.Arazi's truly commonsense approach provides a solid grounding in the subject,explaining principles intuitively from a hardware perspective. He fully covers error correctiontechniques, from basic parity check and single error correction cyclic codes to burst errorcorrecting codes and convolutional codes. All this he presents before introducing Galois fieldtheory - the basic algebraic treatment and theoretical basis of the subject, which usually appearsin the opening chapters of standard textbooks. One entire chapter is devoted to specific practicalissues, such as Reed-Solomon codes (used in compact disc equipment), and maximum length sequences(used in various fields of communications). The basic circuits explained throughout the book areredrawn and analyzed from a theoretical point of view for readers who are interested in tackling themathematics at a more advanced level.Benjamin Arazi is an Associate Professor in the Department ofElectrical and Computer Engineering at the Ben-Gurion University of the Negev. His book is includedin the Computer Systems Series, edited by Herb Schwetman.
Categories: Computers

A First Course in Coding Theory

A First Course in Coding Theory

This book provides and elementary, yet rigorous, introduction to the theory of error-correcting codes.

Author: Raymond Hill

Publisher: Oxford University Press

ISBN: 0198538030

Category: Computers

Page: 251

View: 311

Algebraic coding theory is a new and rapidly developing subject, popular for its many practical applications and for its fascinatingly rich mathematical structure. This book provides an elementary yet rigorous introduction to the theory of error-correcting codes. Based on courses given by the author over several years to advanced undergraduates and first-year graduated students, this guide includes a large number of exercises, all with solutions, making the book highly suitable for individual study.
Categories: Computers

Error Correction Coding and Decoding

Error Correction Coding and Decoding

This book is open access under a CC BY 4.0 license. This book discusses both the theory and practical applications of self-correcting data, commonly known as error-correcting codes.

Author: Martin Tomlinson

Publisher: Springer

ISBN: 9783319511030

Category: Technology & Engineering

Page: 522

View: 533

This book discusses both the theory and practical applications of self-correcting data, commonly known as error-correcting codes. The applications included demonstrate the importance of these codes in a wide range of everyday technologies, from smartphones to secure communications and transactions. Written in a readily understandable style, the book presents the authors’ twenty-five years of research organized into five parts: Part I is concerned with the theoretical performance attainable by using error correcting codes to achieve communications efficiency in digital communications systems. Part II explores the construction of error-correcting codes and explains the different families of codes and how they are designed. Techniques are described for producing the very best codes. Part III addresses the analysis of low-density parity-check (LDPC) codes, primarily to calculate their stopping sets and low-weight codeword spectrum which determines the performance of th ese codes. Part IV deals with decoders designed to realize optimum performance. Part V describes applications which include combined error correction and detection, public key cryptography using Goppa codes, correcting errors in passwords and watermarking. This book is a valuable resource for anyone interested in error-correcting codes and their applications, ranging from non-experts to professionals at the forefront of research in their field. This book is open access under a CC BY 4.0 license.
Categories: Technology & Engineering

A Course in Error correcting Codes

A Course in Error correcting Codes

This book is written as a text for a course aimed at advanced undergraduates.

Author: Jørn Justesen

Publisher: European Mathematical Society

ISBN: 3037190019

Category: Error-correcting codes (Information theory)

Page: 194

View: 840

This book is written as a text for a course aimed at advanced undergraduates. Chapters cover the codes and decoding methods that are currently of most interest in research, development, and application. They give a relatively brief presentation of the essential results, emphasizing the interrelations between different methods and proofs of all important results. A sequence of problems at the end of each chapter serves to review the results and give the student an appreciation of the concepts.
Categories: Error-correcting codes (Information theory)

Error correcting Codes and Finite Fields

Error correcting Codes and Finite Fields

The book assumes only a basic knowledge of linear algebra and develops the mathematical theory in parallel with the codes. Central to the text are worked examples whichmotivate and explain the theory. The book is in four parts.

Author: Oliver Pretzel

Publisher: Oxford University Press on Demand

ISBN: 0192690671

Category: Computers

Page: 341

View: 756

This textbook is a reprint of Chapters 1-20 of the original hardback edition. It provides the reader with the tools necessary to implement modern error-processing schemes. The material on algebraic geometry and geometric Goppa codes, which is not part of a standard introductory course on coding theory, has been omitted. The book assumes only a basic knowledge of linear algebra and develops the mathematical theory in parallel with the codes. Central to the text are worked examples whichmotivate and explain the theory. The book is in four parts. The first introduces the basic ideas of coding theory. The second and third cover the theory of finite fields and give a detailed treatment of BCH and Reed-Solomon codes. These parts are linked by their uses of Eulid's algorithm as a central technique. The fourth part treats classical Goppa codes.
Categories: Computers

Codes for Error Detection

Codes for Error Detection

Only a few older books are devoted to error detecting codes. This book begins with a short introduction to the theory of block codes with emphasis on the parts important for error detection.

Author: Torleiv Klove

Publisher: World Scientific

ISBN: 9789812770516

Category: Computer science

Page: 214

View: 536

There are two basic methods of error control for communication, both involving coding of the messages. With forward error correction, the codes are used to detect and correct errors. In a repeat request system, the codes are used to detect errors and, if there are errors, request a retransmission. Error detection is usually much simpler to implement than error correction and is widely used. However, it is given a very cursory treatment in almost all textbooks on coding theory. Only a few older books are devoted to error detecting codes. This book begins with a short introduction to the theory of block codes with emphasis on the parts important for error detection. The weight distribution is particularly important for this application and is treated in more detail than in most books on error correction. A detailed account of the known results on the probability of undetected error on the q-ary symmetric channel is also given.
Categories: Computer science

List Decoding of Error Correcting Codes

List Decoding of Error Correcting Codes

This question has been investigated extensively starting with the seminal works of Shannon (1948) and Hamming (1950), and has led to the rich theory of “error-correcting codes”.

Author: Venkatesan Guruswami

Publisher: Springer

ISBN: 9783540301806

Category: Computers

Page: 352

View: 751

How can one exchange information e?ectively when the medium of com- nication introduces errors? This question has been investigated extensively starting with the seminal works of Shannon (1948) and Hamming (1950), and has led to the rich theory of “error-correcting codes”. This theory has traditionally gone hand in hand with the algorithmic theory of “decoding” that tackles the problem of recovering from the errors e?ciently. This thesis presents some spectacular new results in the area of decoding algorithms for error-correctingcodes. Speci?cally,itshowshowthenotionof“list-decoding” can be applied to recover from far more errors, for a wide variety of err- correcting codes, than achievable before. A brief bit of background: error-correcting codes are combinatorial str- tures that show how to represent (or “encode”) information so that it is - silient to a moderate number of errors. Speci?cally, an error-correcting code takes a short binary string, called the message, and shows how to transform it into a longer binary string, called the codeword, so that if a small number of bits of the codewordare ?ipped, the resulting string does not look like any other codeword. The maximum number of errorsthat the code is guaranteed to detect, denoted d, is a central parameter in its design. A basic property of such a code is that if the number of errors that occur is known to be smaller than d/2, the message is determined uniquely. This poses a computational problem,calledthedecodingproblem:computethemessagefromacorrupted codeword, when the number of errors is less than d/2.
Categories: Computers

Introduction to Error Control Codes

Introduction to Error Control Codes

Complex mathematical proofs areomitted where possible to keep the text concise and easy-to-follow.Additional notes on the contents:*Chapter 2 the treatment of linear codes in this chapter avoids reference to vector spaces, enabling the ...

Author: Salvatore Gravano

Publisher: Oxford University Press on Demand

ISBN: 0198562314

Category: Computers

Page: 261

View: 886

In the world of digital electronic it is essential to detect and correct errors in digital signals. This introductory book tackles the complex subject of error detection and correction in an easily accessible way for undergraduate students of computer science and electronic engineering. This book leads the reader with no prior knowledge through this important topic while avoiding unnecessary and difficult mathematical proofs.
Categories: Computers