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: 783

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

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: 179

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

Introduction to the Theory of Error Correcting Codes

Introduction to the Theory of Error Correcting Codes

This new edition features: * A greater emphasis on nonlinear binary codes * An exciting new discussion on the relationship between codes and combinatorial games * Updated and expanded sections on the Vashamov-Gilbert bound, van Lint-Wilson ...

Author: Vera Pless

Publisher: John Wiley & Sons

ISBN: 9781118030998

Category: Mathematics

Page: 224

View: 609

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.
Categories: Mathematics

Error Correcting Codes

Error Correcting Codes

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 ...

Author: D J. Baylis

Publisher: Routledge

ISBN: 9781351449830

Category: Mathematics

Page: 232

View: 703

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

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

ISBN: 0792390172

Category: Technology & Engineering

Page: 289

View: 865

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

Foundations of Coding

Foundations of Coding

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

Author: Jiri Adamek

Publisher: John Wiley & Sons

ISBN: 9781118031513

Category: Computers

Page: 352

View: 141

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.
Categories: Computers

Introduction to Error Control Codes

Introduction to Error Control Codes

This book leads the reader with no prior knowledge through this important topic while avoiding unnecessary and difficult mathematical proofs.

Author: Salvatore Gravano

Publisher: Oxford University Press on Demand

ISBN: UOM:39015051283300

Category: Computers

Page: 261

View: 998

This textbook provides a firm foundation to the field of error control codes, leading the student step by step through this complex topic, beginning with single parity code checks and repetition codes. Through these basic error-control mechanisms the fundamental principles of error detectionand correction, minimum distance and error-control limits are considered. The reader is guided from basic error-control codes through to linear codes, cyclic codes, linear feedback shift registers, vector fields, Galois fields, BCH codes and convolutional 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 reader to gain an understanding of linear codes sufficient to move onto cyclic codes without the distraction of the complexity of vector spaces.*Chapter 5 considers vector spaces and revisits linear codes in the context of vector spaces.*Chapter 6 starts with the ordinary concepts of sets and gently leads the reader through the principles of groups and fields, the end goal being an understanding of Galois fields that will enable the reader to understand the BCH codes considered in Chapter 7.
Categories: Computers

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: 489

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

Introduction to Error correcting Codes

Introduction to Error correcting Codes

This title covers the fundamentals of error-correcting codes for the non-specialist user, from general concepts through to the most common error-correcting codes applied to bit transmission in packet switched networks and personal cellular ...

Author: Michael Purser

Publisher: Artech House on Demand

ISBN: 0890067848

Category: Technology & Engineering

Page: 133

View: 503

This title covers the fundamentals of error-correcting codes for the non-specialist user, from general concepts through to the most common error-correcting codes applied to bit transmission in packet switched networks and personal cellular systems. It also includes sections on linear blockades, BCH codes and cyclic codes.
Categories: Technology & Engineering

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: 102

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 Correction Coding

Error Correction Coding

This is supplemented by a lab-oriented approach, in which readers are provided tools to implement encoding and decoding algorithms with explicit algorithm statements. Block codes and convolutional codes are discussed.

Author: Todd K. Moon

Publisher: John Wiley & Sons

ISBN: 9781119567479

Category: Computers

Page: 992

View: 461

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.
Categories: Computers

Author: 内田興二

Publisher:

ISBN: OCLC:166456398

Category:

Page: 110

View: 334

Categories:

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: 9781139439503

Category: Technology & Engineering

Page:

View: 819

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

Introduction to Coding Theory

Introduction to Coding Theory

Mastering the contents of this book brings a complete understanding of the theory of cyclic codes, including their various applications and the Euclidean algorithm decoding of BCH-codes, and carries readers to the level of the most recent ...

Author: Jurgen Bierbrauer

Publisher: CRC Press

ISBN: 9781351989602

Category: Mathematics

Page: 390

View: 273

Although its roots lie in information theory, the applications of coding theory now extend to statistics, cryptography, and many areas of pure mathematics, as well as pervading large parts of theoretical computer science, from universal hashing to numerical integration. Introduction to Coding Theory introduces the theory of error-correcting codes in a thorough but gentle presentation. Part I begins with basic concepts, then builds from binary linear codes and Reed-Solomon codes to universal hashing, asymptotic results, and 3-dimensional codes. Part II emphasizes cyclic codes, applications, and the geometric desciption of codes. The author takes a unique, more natural approach to cyclic codes that is not couched in ring theory but by virtue of its simplicity, leads to far-reaching generalizations. Throughout the book, his discussions are packed with applications that include, but reach well beyond, data transmission, with each one introduced as soon as the codes are developed. Although designed as an undergraduate text with myriad exercises, lists of key topics, and chapter summaries, Introduction to Coding Theory explores enough advanced topics to hold equal value as a graduate text and professional reference. Mastering the contents of this book brings a complete understanding of the theory of cyclic codes, including their various applications and the Euclidean algorithm decoding of BCH-codes, and carries readers to the level of the most recent research.
Categories: Mathematics

Block Error Correcting Codes

Block Error Correcting Codes

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.

Author: Sebastian Xambo-Descamps

Publisher: Springer Science & Business Media

ISBN: 3540003959

Category: Computers

Page: 266

View: 625

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.
Categories: Computers

Error Correction Codes for Non Volatile Memories

Error Correction Codes for Non Volatile Memories

O. Papini, “Algebre discrete et codes correcteurs”, Springer-Verlag, 1995. W. W. Peterson, E. J. Weldon, “Error-Correcting Codes”, MIT Press, 1998. V. Pless, “Introduction to the Theory of Error-Correcting Codes”, John Wiley and Sons, ...

Author: Rino Micheloni

Publisher: Springer Science & Business Media

ISBN: 9781402083914

Category: Technology & Engineering

Page: 338

View: 374

Nowadays it is hard to find an electronic device which does not use codes: for example, we listen to music via heavily encoded audio CD's and we watch movies via encoded DVD's. There is at least one area where the use of encoding/decoding is not so developed, yet: Flash non-volatile memories. Flash memory high-density, low power, cost effectiveness, and scalable design make it an ideal choice to fuel the explosion of multimedia products, like USB keys, MP3 players, digital cameras and solid-state disk. In ECC for Non-Volatile Memories the authors expose the basics of coding theory needed to understand the application to memories, as well as the relevant design topics, with reference to both NOR and NAND Flash architectures. A collection of software routines is also included for better understanding. The authors form a research group (now at Qimonda) which is the typical example of a fruitful collaboration between mathematicians and engineers.
Categories: Technology & Engineering

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: Birkhäuser

ISBN: 3030411524

Category: Mathematics

Page: 180

View: 410

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

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: 992

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