Verfügbarkeit: Algebraic coding theory | Technische Universität München

Algebraic coding theory:

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Bibliographische Detailangaben
Beteilige Person:	Berlekamp, Elwyn R. 1940-2019 (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Singapore World Scientific 2015
Ausgabe:	Revised edition
Schlagwörter:	Codierung Codierungstheorie Algebraische Codierung Informationstheorie
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028045398&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028045398&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA
Beschreibung:	Includes bibliographical references and index
Umfang:	xxiv, 474 pages
ISBN:	9789814635905 9814635901 9789814635899 9814635898

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Datensatz im Suchindex

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adam_text	rfc/n Algebraic Coding Theory In 1948, Claude Shannon showed the existence of codes which could correct errors. Over the next two decades, he and others explored the theoretical limits of the performance of long block codes, resulting in curves such as the one shown above. In the 1960s, mathematicians introduced the notion that the sophisticated algebra of finite fields might be used to design and implement such codes. When the first edition of this book appeared in 1968, it immediately became a landmark in the field. It introduced a novel algorithm for factoring polynomials, the crux of which is the top formula on the left of the cover. It also introduced a new algorithm for determining the polynomial that needs to be factored in order to decode Reed-Solomon codes. The crux of this algorithm, which appears on the bottom left of the cover, became widely known as the Berlekamp-Massey algorithm. These advances made algebraic error-correcting codes feasible for a wide range of applications in both computer memories and in digital communications. Building on the foundations of this book, a subsequent breakthrough in the 1970s reduced the complexity of nonbinary Reed-Solomon encoders to that of much simpler binary codes of comparable redundancies. A circuit diagram for such an encoder appears on the right of the cover. It became a NASA standard for deep space communications. ISBN 978-981 -4635-89-9 9II; ^898 14 635 ► 899 World Scientific www.worldscientific.com 9407 he CONTENTS R Preface to the Revised Edition vii Preface to the Second Edition xiii Preface xv Acknowledgments xix 1. Basic Binary Codes 1 1.1 Repetition Codes and Singie-Parity-Check Codes 1 1.2 Linear Codes 4 1.3 Hamming Codes 8 1.4 Manipulative Introduction to Double-Error-Correcting BCH Codes 12 Problems 20 2. Arithmetic Operations Modulo an Irreducible Binary Polynomial 21 2.1 A Closer Look at Euclid’s Algorithm 21 2.2 Logical Circuitry 30 2.3 Multiplicative Inversion 36 2.4 Multiplication 44 2.5 The Solution of Simultaneous Linear Equations 51 2.6 Special Methods for Solving Simultaneous Linear Equations When the Coefficient Matrix is Mostly Zeros 63 Problems 69 3. The Number of Irreducible g-ary Polynomials of Given Degree 70 3.1 A Brute-Force Attack 70 3.2 Generating Functions 73 3.3 The Number of Irreducible Monic g-ary Polynomials of Given Degree—A Refined Approach 76 3.4 The Moebius Inversion Formulas 81 Problems 85 * STARRED SECTIONS MAY BE SKIPPED ON FIRST READING. XXI XXII CONTENTS 4. The Structure of Finite Fields 87 4.1 Definitions 87 4.2 Multiplicative Structure of Finite Fields 88 4.3 Cyclotomie Polynomials 90 4.4 Algebraic Structure of Finite Fields 96 4.5 Examples 105 ★4.6 Algebraic Closure 111 ★4.7 Determining Minimal Polynomials 112 Problems 117 5. Cyclic Binary Codes 119 5.1 Reordering the Columns of the Parity-Check Matrix of Hamming Codes 119 5.2 Reordering the Columns of the Parity-Check Matrix of Double-Error-Correcting Binary BCH Codes 125 5.3 General Properties of Cyclic Codes 129 5.4 The Chien Search 132 5.5 Outline of General Decoder for Any Cyclic Binary Code 136 5.6 Example 138 5.7 Example 139 5.8 Equivalence of Cyclic Codes Defined in Terms of Different Primitive nth Roots of Unity 141 Problems 144 6. The Factorization of Polynomials Over Finite Fields 146 6.1 A General Algorithm 146 ★6.2 Determining the Period of a Polynomial 150 ★6.3 Trinomials Over GF(2) 153 6.4 Factoring xn — 1 Explicitly 154 ★6.5 Determining the Degrees of the irreducible Factors of the Cyclotomie Polynomials 156 ★6.6 Is the Number of Irreducible Factors of f(x) Over GF(q) Odd or Even? 159 ★6.7 Quadratic Reciprocity 171 Problems 173 7. Binary BCH Codes for Correcting Multiple Errors 176 7.1 Examples 176 7.2 The Key Equation for Decoding Binary BCH Codes 178 7.3 Heuristic Solution of the Key Equation 180 7.4 An Algorithm for Solving the Key Equation Over Any Field 184 ★7.5 Relation to Matrix Decoding Methods 188 ★7.6 Simplifications in Algorithm 7.4 for Binary BCH Codes 192 7.7 Implementation of Binary BCH Decoders 195 Problems 199 CONTENTS xxiii 8. Nonbinary Coding 200 8.1 Modulation Schemes 200 8.2 Weight Functions 204 Problem 206 9. Negacyclic Codes for the Lee Metric 207 9.1 Error Locations and the Error-Locator Polynomial 207 9.2 Double-Error-Correcting Codes 209 9.3 Negacyclic Codes 211 Problems 217 10. Gorenstein-Zierler Generalized Nonbinary BCH Codes for the Hamming Metric 218 10.1 The Generalized BCH Codes and An Algorithm to Decode Them 218 10.2 Examples 221 10.3 Alternate BCH Codes and Extended BCH Codes 223 10.4 Decoding Erasures as Well as Errors 229 10.5 Decoding More Than tErrors 231 10.6 Examples 237 Problem 240 11. Linearized Polynomials and Affine Polynomials 241 11.1 How to Find Their Roots 241 11.2 The Least Affine Multiple 245 ★11.3 Abstract Properties of Linearized and Affine Polynomials 247 ★11.4 Transformations of f(z) 255 ★11.5 Root Counting 257 ★11.6 Low-Weight Codewords in Certain Codes 262 Problems 271 12. The Enumeration of Information Symbols in BCH Codes 273 12.1 Converting the Problem to an Enumeration of Certain Integers mod n 273 ★12.2 Converting the Problem for Primitive BCH Codes to an Enumeration of Certain g-ary Sequences 275 ★12.3 Sequence Enumeration Theorems 278 ★12.4 Examples 284 ★12.5 The Enumeration of Information Symbols in Nonprimitive BCH Codes 287 12.6 Asymptotic Results 290 ★12.7 Actual Distance 294 Problems 296 13. The Information Rate of the Optimum Codes 298 13.1 The Hamming-Rao High-Rate Volume Bound 298 13.2 Perfect Codes 302 13.3 The Bound d n + 1 — fc 309 13.4 Plotkin’s Low-Rate Average Distance Bound 311 13.5 Equidistant Codes 315 13.6 The Elias Bound 318 xxiv 13.7 The Gilbert Bound 13.8 Asymptotic Error Bounds and Finite Special Cases 14. Codes Derived by Modifying or Combining Other Codes 14.1 Extending a Code by Annexing More Check Digits 14.2 Puncturing a Code by Deleting Check Digits 14.3 Augmenting Additional Codewords to the Code 14.4 Expurgating Codewords from the Code 14.5 Lengthening the Code by Annexing More Message Digits 14.6 Shortening the Code by Omitting Message Digits 14.7 Subfield Subcodes 14.8 Direct-Product Codes and Their Relatives 14.9 Concatenated Codes 15. Other Important Coding and Decoding Methods 15.1 Srivastava Codes—Noncyclic Codes with an Algebraic Decoding Algorithm 15.2 Residue Codes—Good Codes Which are Hard to Decode 15.3 Reed-Muller Codes—Weak Codes Which are Easy to Decode 15.4 Threshold Decoding—The Best Known Algorithm for Decoding Certain Codes 15.5 Orthogonalizable Codes Based on Finite Geometries 15.6 Convolutional Codes—A Survey Problems 16. Weight Enumerators 16.1 The Relationship between Weight Enumerators and the Probability of Decoding Failure 16.2 The MacWilliams-Pless Equations for the Weight Enumerators of Dual Codes ★16.3 Weight Restrictions ★16.4 Kasami’s Weight Enumerators for Certain Subcodes of the Second-Order RM Code ★16.5 The Weight Enumerator for the Reed-Solomon Codes Appendix A Appendix B References References to the Second Edition Index CONTENTS 321 324 331 333 334 335 335 336 336 336 338 347 350 350 352 361 367 375 388 394 397 397 400 407 417 429 441 442 443 453 461
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spellingShingle	Berlekamp, Elwyn R. 1940-2019 Algebraic coding theory Codierung (DE-588)4070059-8 gnd Codierungstheorie (DE-588)4139405-7 gnd Algebraische Codierung (DE-588)4141834-7 gnd Informationstheorie (DE-588)4026927-9 gnd
subject_GND	(DE-588)4070059-8 (DE-588)4139405-7 (DE-588)4141834-7 (DE-588)4026927-9
title	Algebraic coding theory
title_auth	Algebraic coding theory
title_exact_search	Algebraic coding theory
title_full	Algebraic coding theory Elwyn Berlekamp
title_fullStr	Algebraic coding theory Elwyn Berlekamp
title_full_unstemmed	Algebraic coding theory Elwyn Berlekamp
title_short	Algebraic coding theory
title_sort	algebraic coding theory
topic	Codierung (DE-588)4070059-8 gnd Codierungstheorie (DE-588)4139405-7 gnd Algebraische Codierung (DE-588)4141834-7 gnd Informationstheorie (DE-588)4026927-9 gnd
topic_facet	Codierung Codierungstheorie Algebraische Codierung Informationstheorie
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