We have gathered linear codes in classes of codes which are of the same quality with respect to error correction. Since the metric structure of a code determines its error correction properties we have introduced the notion of isometric ...
Author: Anton Betten
Publisher: Springer Science & Business Media
ISBN: 9783540317036
Category: Mathematics
Page: 798
View: 353
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.
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: 162
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.
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: 702
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.
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: 207
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.
These codes are called trivial perfect codes. ◇ Exercise 80 Prove that a perfect t-error-correcting linear code of length n has precisely ( n i ) cosets of weight i for 0 ≤ i ≤ t and no other cosets. Hint: How many weight i vectors ...
Author: W. Cary Huffman
Publisher: Cambridge University Press
ISBN: 9781139439503
Category: Technology & Engineering
Page:
View: 794
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.
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 ...
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.
But from the proof of Theorem 6.1 this n is precisely the maximal number of columns of length n - I with no pair of columns dependent. Hence, by definition C is a Hamming code. Moving on to 2-error correcting linear codes, the condition ...
Author: D J. Baylis
Publisher: Routledge
ISBN: 9781351449847
Category: Mathematics
Page: 232
View: 373
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.
This book starts by establishing the basic linear network error correction (LNEC) model and then characterizes two equivalent descriptions.
Author: Xuan Guang
Publisher: Springer Science & Business Media
ISBN: 9781493905881
Category: Computers
Page: 107
View: 570
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.
Corollary 10.34 tells us that uniformly packed e-error correcting binary codes hold t-designs. However, we do not have results on possible t-designs held in uniformly packed e-error correcting codes over GF(q) for q > 2.
Author: Cunsheng Ding
Publisher: World Scientific
ISBN: 9789811251344
Category: Computers
Page: 540
View: 949
Since the publication of the first edition of this monograph, a generalisation of the Assmus-Mattson theorem for linear codes over finite fields has been developed, two 70-year breakthroughs and a considerable amount of other progress on t-designs from linear codes have been made. This second edition is a substantial revision and expansion of the first edition. Two new chapters and two new appendices have been added, and most chapters of the first edition have been revised.It provides a well-rounded and detailed account of t-designs from linear codes. Most chapters of this book cover the support designs of linear codes. A few chapters deal with designs obtained from linear codes in other ways. Connections among ovals, hyperovals, maximal arcs, ovoids, special functions, linear codes and designs are also investigated. This book consists of both classical and recent results on designs from linear codes.It is intended to be a reference for postgraduates and researchers who work on combinatorics, or coding theory, or digital communications, or finite geometry. It can also be used as a textbook for postgraduates in these subject areas.
Author: Florence Jessie MacWilliamsPublish On: 1977-01-01
Starting with the (8, 20, 3), . . . , (ll, 144, 3) codes given in §7, we can construct an infinite family: '€, ... It follows from the sphere-packing bound (Theorem 6 of Ch. 1) that the largest single-error-correcting linear code of the ...