# Error Correcting Linear Codes

In the third section we will investigate how to detect and correct burst errors from
a general point of view. We have already mentioned that in fact two linear codes
are applied for the encoding process in the production of a compact disc.

Author: Anton Betten

Publisher: Springer Science & Business Media

ISBN: 9783540317036

Category: Mathematics

Page: 798

View: 902

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

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: 947

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

# 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: 120

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

# 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: 605

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

# An Introduction to Error Correcting Codes with Applications

codes 180 - for first-order Reed-Muller codes 125 - for linear codes with both
errors and erasures 244 - for single-error-correcting linear codes 68 - standard
array decoding for linear codes 75 - syndrome decoding for linear codes 77 ...

Author: Scott A. Vanstone

Publisher: Springer Science & Business Media

ISBN: 9781475720327

Category: Technology & Engineering

Page: 289

View: 238

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

# A Course in Error correcting Codes

The discussion of error mechanisms and channel models is postponed to
Chapter 4 and here we will simply consider the number of errors that the code
can correct . One of the most important classes of codes , the linear block codes ,
is ...

Author: Jørn Justesen

Publisher: European Mathematical Society

ISBN: 3037190019

Category: Error-correcting codes (Information theory)

Page: 194

View: 686

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)

# Block Error Correcting Codes

1 Block Error-correcting Codes Channel coding is a very young field. However, it
has gained importance in ... In the second section, we look at linear codes, also
from a general point of view. We include a presentation of syndrome decoding.

Author: Sebastian Xambo-Descamps

Publisher: Springer Science & Business Media

ISBN: 9783642189975

Category: Computers

Page: 266

View: 326

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

# 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: 383

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

# The Theory of Error Correcting Codes

Author: Florence Jessie MacWilliams

Publisher: Elsevier

ISBN: 9780444850102

Category: Electronic books

Page: 762

View: 218

Categories: Electronic books

# List Decoding of Error Correcting Codes

These linear-time codes significantly improve the fraction of errors corrected by
the earlier linear-time codes due to Spielman [176]. Our codes are obtained by
using Spielman's codes as a building block and then boosting its error-resilience
...

Author: Venkatesan Guruswami

Publisher: Springer Science & Business Media

ISBN: 9783540240518

Category: Computers

Page: 350

View: 519

This monograph is a thoroughly revised and extended version of the author's PhD thesis, which was selected as the winning thesis of the 2002 ACM Doctoral Dissertation Competition. Venkatesan Guruswami did his PhD work at the MIT with Madhu Sudan as thesis adviser. Starting with the seminal work of Shannon and Hamming, coding theory has generated a 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 transmission errors efficiently. This book presents some spectacular new results in the area of decoding algorithms for error-correcting codes. Specificially, it shows how the notion of list-decoding can be applied to recover from far more errors, for a wide variety of error-correcting codes, than achievable before The style of the exposition is crisp and the enormous amount of information on combinatorial results, polynomial time list decoding algorithms, and applications is presented in well structured form.
Categories: Computers

# A Short Course on Error Correcting Codes

2 Linear Codes § 2.1 Introduction To make, codes easier to use and to analyze
we must impose some algebraic structure on them. The simplest assumption is
that the code is linear. In this chapter we give the basic theory of linear codes, ...

Author: N.J.A. Sloane

Publisher: Springer

ISBN: 9783709128640

Category: Computers

Page: 78

View: 339

Categories: Computers

# Applied Algebra Algebraic Algorithms and Error Correcting Codes

Linear Codes and Polylinear Recurrences over Finite Rings and Modules (A
Survey) V.L. Kurakin, A.S. Kuzmin, V.T. Markov, A.V. Mikhalev, and A.A. Nechaev
Department of Mechanics and Mathematics and Center of New Information ...

Author: Marc Fossorier

Publisher: Springer Science & Business Media

ISBN: 9783540667230

Category: Computers

Page: 510

View: 911

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.
Categories: Computers

# Linear Network Error Correction Coding

Hence, the performance analysis of random linear network coding is important in
theory and application. It is characterized by widely studying the different failure
probabilities including failure probability at sink node, failure probability for ...

Author: Xuan Guang

Publisher: Springer Science & Business Media

ISBN: 9781493905881

Category: Computers

Page: 107

View: 174

There are two main approaches in the theory of network error correction coding. In this SpringerBrief, the authors summarize some of the most important contributions following the classic approach, which represents messages by sequences similar to algebraic coding, and also briefly discuss the main results following the other approach, that uses the theory of rank metric codes for network error correction of representing messages by subspaces. This book starts by establishing the basic linear network error correction (LNEC) model and then characterizes two equivalent descriptions. Distances and weights are defined in order to characterize the discrepancy of these two vectors and to measure the seriousness of errors. Similar to classical error-correcting codes, the authors also apply the minimum distance decoding principle to LNEC codes at each sink node, but use distinct distances. For this decoding principle, it is shown that the minimum distance of a LNEC code at each sink node can fully characterize its error-detecting, error-correcting and erasure-error-correcting capabilities with respect to the sink node. In addition, some important and useful coding bounds in classical coding theory are generalized to linear network error correction coding, including the Hamming bound, the Gilbert-Varshamov bound and the Singleton bound. Several constructive algorithms of LNEC codes are presented, particularly for LNEC MDS codes, along with an analysis of their performance. Random linear network error correction coding is feasible for noncoherent networks with errors. Its performance is investigated by estimating upper bounds on some failure probabilities by analyzing the information transmission and error correction. Finally, the basic theory of subspace codes is introduced including the encoding and decoding principle as well as the channel model, the bounds on subspace codes, code construction and decoding algorithms.
Categories: Computers

# 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: 809

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

# Applied Algebra Algebraic Algorithms and Error Correcting Codes

16th International Symposium, AAECC-16, Las Vegas, NV, USA, February 20-24,
2006, Proceedings Marc Fossorier, Hideki Imai, Shu Lin, Alain Poli. A General
Framework for Applying FGLM Techniques to Linear Codes M. Borges-
Quintana1 ...

Author: Marc Fossorier

Publisher: Springer Science & Business Media

ISBN: 9783540314233

Category: Computers

Page: 337

View: 892

This book constitutes the refereed proceedings of the 16th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-16, held in Las Vegas, NV, USA in February 2006. The 25 revised full papers presented together with 7 invited papers were carefully reviewed and selected from 32 submissions. Among the subjects addressed are block codes; algebra and codes: rings, fields, and AG codes; cryptography; sequences; decoding algorithms; and algebra: constructions in algebra, Galois groups, differential algebra, and polynomials.
Categories: Computers

# Error Correction Coding for Digital Communications

As we will discover subsequently, this property has two far-reaching effects. The
first is that it vastly simplifies the encoding and decoding problem by allowing one
to express any code word as the "linear" combination of a small set of reference ...

Author: George C. Clark Jr.

Publisher: Springer Science & Business Media

ISBN: 0306406152

Category: Technology & Engineering

Page: 422

View: 473

Error-correction coding is being used on an almost routine basis in most new communication systems. Not only is coding equipment being used to increase the energy efficiency of communication links, but coding ideas are also providing innovative solutions to many related communication problems. Among these are the elimination of intersymbol interference caused by filtering and multipath and the improved demodulation of certain frequency modulated signals by taking advantage of the "natural" coding provided by a continuous phase. Although several books and nu merous articles have been written on coding theory, there are still noticeable deficiencies. First, the practical aspects of translating a specific decoding algorithm into actual hardware have been largely ignored. The information that is available is sketchy and is widely dispersed. Second, the information required to evaluate a particular technique under situations that are en countered in practice is available for the most part only in private company reports. This book is aimed at correcting both of these problems. It is written for the design engineer who must build the coding and decoding equipment and for the communication system engineer who must incorporate this equipment into a system. It is also suitable as a senior-level or first-year graduate text for an introductory one-semester course in coding theory. The book U"Ses a minimum of mathematics and entirely avoids the classical theorem/proof approach that is often seen in coding texts.
Categories: Technology & Engineering

# Error Correcting Coding and Security for Data Networks

In accordance with this criterion a linear code is capable of correcting a single
error if, and only if, a parity-check matrix H of this code does not contain collinear
columns. In a binary case that means matrix H consists of distinct nonzero
columns ...

Author: Grigorii Kabatiansky

Publisher: John Wiley & Sons

ISBN: 9780470867563

Category: Technology & Engineering

Page: 288

View: 509

Error correcting coding is often analyzed in terms of its application to the separate levels within the data network in isolation from each other. In this fresh approach, the authors consider the data network as a superchannel (a multi-layered entity) which allows error correcting coding to be evaluated as it is applied to a number of network layers as a whole. By exposing the problems of applying error correcting coding in data networks, and by discussing coding theory and its applications, this original technique shows how to correct errors in the network through joint coding at different network layers. Discusses the problem of reconciling coding applied to different layers using a superchannel approach Includes thorough coverage of all the key codes: linear block codes, Hamming, BCH and Reed-Solomon codes, LDPC codes decoding, as well as convolutional, turbo and iterative coding Considers new areas of application of error correcting codes such as transport coding, code-based cryptosystems and coding for image compression Demonstrates how to use error correcting coding to control such important data characteristics as mean message delay Provides theoretical explanations backed up by numerous real-world examples and practical recommendations Features a companion website containing additional research results including new constructions of LDPC codes, joint error-control coding and synchronization, Reed-Muller codes and their list decoding By progressing from theory through to practical problem solving, this resource contains invaluable advice for researchers, postgraduate students, engineers and computer scientists interested in data communications and applications of coding theory.
Categories: Technology & Engineering

# Applied Algebra Algebraic Algorithms and Error Correcting Codes

In addition, double circulant codes with the largest minimum Lee weights for this
class of codes are presented for lengths up to 32. 1 Introduction Some of the best
known non-linear binary codes which are better than any comparable linear ...

Author: Serdar Boztas

Publisher: Springer Science & Business Media

ISBN: 9783540429111

Category: Mathematics

Page: 404

View: 433

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.
Categories: Mathematics

# 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: 864

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

# Polynomial Theory of Error Correcting Codes

3.1 Traditional View of Non-systematic s.s. Time-Invariant Convolutional Codes
In the previous chapters, the following conclusion has been reached: starting
from a code with minimum distance d > 2, in order to have a linear growth in the ...

Author: Giovanni Cancellieri

Publisher: Springer

ISBN: 9783319017273

Category: Technology & Engineering

Page: 732

View: 781

The book offers an original view on channel coding, based on a unitary approach to block and convolutional codes for error correction. It presents both new concepts and new families of codes. For example, lengthened and modified lengthened cyclic codes are introduced as a bridge towards time-invariant convolutional codes and their extension to time-varying versions. The novel families of codes include turbo codes and low-density parity check (LDPC) codes, the features of which are justified from the structural properties of the component codes. Design procedures for regular LDPC codes are proposed, supported by the presented theory. Quasi-cyclic LDPC codes, in block or convolutional form, represent one of the most original contributions of the book. The use of more than 100 examples allows the reader gradually to gain an understanding of the theory, and the provision of a list of more than 150 definitions, indexed at the end of the book, permits rapid location of sought information.
Categories: Technology & Engineering