Introduction to coding theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2007
|
| Edition: | 1. publ., repr. with corr. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 566 S. graph. Darst. |
| ISBN: | 0521845041 9780521845045 |
Staff View
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| adam_text | Contents Preface page ix 1 Introduction 1.1 Communication systems 1.2 Channel coding 1.3 Block codes 1.4 Decoding 1.5 Levels of error handling Problems Notes 1 1 3 5 7 11 17 22 2 Linear Codes 2.1 Definition 2.2 Encoding of linear codes 2.3 Parity-check matrix 2.4 Decoding of linear codes Problems Notes 26 26 28 29 32 36 47 3 Introduction to Finite Fields 3.1 Prime fields 3.2 Polynomials 3.3 Extension fields 3.4 Roots of polynomials 3.5 Primitive elements 3.6 Field characteristic 3.7 Splitting field 3.8 Application: double error-correcting codes Problems Notes 50 50 51 56 59 60 62 64 66 70 90 v
Contents VI Contents vu Bounds on the Parameters of Codes 4.1 The Singleton bound 4.2 The sphere-packing bound 4.3 The Gilbert-Varshamov bound 4.4 MacWilliams’ identities 4.5 Asymptotic bounds 4.6 Converse Coding Theorem 4.7 Coding Theorem Problems Notes 93 94 95 97 99 104 110 115 119 136 5 Reed-Solomon andRelated Codes 5.1 Generalized Reed-Solomon codes 5.2 Conventional Reed-Solomon codes 5.3 Encoding of RS codes 5.4 Concatenated codes 5.5 Alternant codes 5.6 BCH codes Problems Notes 147 148 151 152 154 157 162 163 177 6 Decoding of Reed-Solomon Codes 6.1 Introduction 6.2 Syndrome computation 6.3 Key equation of GRS decoding 6.4 Solving the key equation by Euclid’s algorithm 6.5 Finding the error values 6.6 Summary of the GRS decoding algorithm 6.7 The Berlekamp-Massey algorithm Problems Notes 183 183 184 185 191 194 195 197 204 215 10 Codes in the Lee Metric 10.1 Lee weight and Lee distance 10.2 Newton’s identities 10.3 Lee-metric alternant codes and GRS codes 10.4 Decoding alternant codes in the Lee metric 10.5 Decoding GRS codes in the Lee metric 10.6 Berlekamp codes 10.7 Bounds for codes in the Lee metric Problems Notes 298 298 300 302 306 312 314 316 321 327 7 Structure of FiniteFields 7.1 Minimal polynomials 7.2 Enumeration of irreducible polynomials 7.3 Isomorphism of finite fields 7.4 Primitive polynomials 7.5 Cyclotomie cosets Problems Notes 218 218 224 227 227 229 232 240 11 MDS Codes 11.1 Definition revisited 11.2 GRS codes and their extensions 11.3 Bounds on the length of linear MDS codes 11.4 GRS codes and the MDS conjecture 11.5 Uniqueness of certain MDS
codes Problems Notes 333 333 335 338 342 347 351 361 4 8 Cyclic Codes 8.1 Definition 8.2 Generator polynomial and check polynomial 8.3 Roots of a cyclic code 8.4 BCH codes as cyclic codes 8.5 The BCH bound Problems Notes 242 242 244 247 250 253 256 265 9 List Decoding of Reed-Solomon Codes 9.1 List decoding 9.2 Bivariate polynomials 9.3 GRS decoding through bivariate polynomials 9.4 Sudan’s algorithm 9.5 The Guruswami-Sudan algorithm 9.6 List decoding of alternant codes 9.7 Finding linear bivariate factors 9.8 Bounds on the decoding radius Problems Notes 266 267 268 269 271 276 280 284 289 291 295
viii Contents 12 Concatenated Codes 12.1 Definition revisited 12.2 Decoding of concatenated codes 12.3 The Zyablov bound 12.4 Justesen codes 12.5 Concatenated codes that attain capacity Problems Notes 365 366 367 371 374 378 381 392 13 Graph Codes 13.1 Basic concepts from graph theory 13.2 Regular graphs 13.3 Graph expansion 13.4 Expanders from codes 13.5 Ramanujan graphs 13.6 Codes from expanders 13.7 Iterative decoding of graph codes 13.8 Graph codes in concatenated schemes Problems Notes 395 396 401 402 406 409 411 414 420 426 445 14 Trellis and Convolutional Codes 14.1 Labeled directed graphs 14.2 Trellis codes 14.3 Decoding of trellis codes 14.4 Linear finite-state machines 14.5 Convolutional codes 14.6 Encoding of convolutional codes 14.7 Decoding of convolutional codes 14.8 Non-catastrophic generator matrices Problems Notes 452 453 460 466 471 477 479 485 495 501 518 Appendix: Basics inModern Algebra Problems 521 522 Bibliography 527 List of Symbols 553 Index 559
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| adam_txt |
Contents Preface page ix 1 Introduction 1.1 Communication systems 1.2 Channel coding 1.3 Block codes 1.4 Decoding 1.5 Levels of error handling Problems Notes 1 1 3 5 7 11 17 22 2 Linear Codes 2.1 Definition 2.2 Encoding of linear codes 2.3 Parity-check matrix 2.4 Decoding of linear codes Problems Notes 26 26 28 29 32 36 47 3 Introduction to Finite Fields 3.1 Prime fields 3.2 Polynomials 3.3 Extension fields 3.4 Roots of polynomials 3.5 Primitive elements 3.6 Field characteristic 3.7 Splitting field 3.8 Application: double error-correcting codes Problems Notes 50 50 51 56 59 60 62 64 66 70 90 v
Contents VI Contents vu Bounds on the Parameters of Codes 4.1 The Singleton bound 4.2 The sphere-packing bound 4.3 The Gilbert-Varshamov bound 4.4 MacWilliams’ identities 4.5 Asymptotic bounds 4.6 Converse Coding Theorem 4.7 Coding Theorem Problems Notes 93 94 95 97 99 104 110 115 119 136 5 Reed-Solomon andRelated Codes 5.1 Generalized Reed-Solomon codes 5.2 Conventional Reed-Solomon codes 5.3 Encoding of RS codes 5.4 Concatenated codes 5.5 Alternant codes 5.6 BCH codes Problems Notes 147 148 151 152 154 157 162 163 177 6 Decoding of Reed-Solomon Codes 6.1 Introduction 6.2 Syndrome computation 6.3 Key equation of GRS decoding 6.4 Solving the key equation by Euclid’s algorithm 6.5 Finding the error values 6.6 Summary of the GRS decoding algorithm 6.7 The Berlekamp-Massey algorithm Problems Notes 183 183 184 185 191 194 195 197 204 215 10 Codes in the Lee Metric 10.1 Lee weight and Lee distance 10.2 Newton’s identities 10.3 Lee-metric alternant codes and GRS codes 10.4 Decoding alternant codes in the Lee metric 10.5 Decoding GRS codes in the Lee metric 10.6 Berlekamp codes 10.7 Bounds for codes in the Lee metric Problems Notes 298 298 300 302 306 312 314 316 321 327 7 Structure of FiniteFields 7.1 Minimal polynomials 7.2 Enumeration of irreducible polynomials 7.3 Isomorphism of finite fields 7.4 Primitive polynomials 7.5 Cyclotomie cosets Problems Notes 218 218 224 227 227 229 232 240 11 MDS Codes 11.1 Definition revisited 11.2 GRS codes and their extensions 11.3 Bounds on the length of linear MDS codes 11.4 GRS codes and the MDS conjecture 11.5 Uniqueness of certain MDS
codes Problems Notes 333 333 335 338 342 347 351 361 4 8 Cyclic Codes 8.1 Definition 8.2 Generator polynomial and check polynomial 8.3 Roots of a cyclic code 8.4 BCH codes as cyclic codes 8.5 The BCH bound Problems Notes 242 242 244 247 250 253 256 265 9 List Decoding of Reed-Solomon Codes 9.1 List decoding 9.2 Bivariate polynomials 9.3 GRS decoding through bivariate polynomials 9.4 Sudan’s algorithm 9.5 The Guruswami-Sudan algorithm 9.6 List decoding of alternant codes 9.7 Finding linear bivariate factors 9.8 Bounds on the decoding radius Problems Notes 266 267 268 269 271 276 280 284 289 291 295
viii Contents 12 Concatenated Codes 12.1 Definition revisited 12.2 Decoding of concatenated codes 12.3 The Zyablov bound 12.4 Justesen codes 12.5 Concatenated codes that attain capacity Problems Notes 365 366 367 371 374 378 381 392 13 Graph Codes 13.1 Basic concepts from graph theory 13.2 Regular graphs 13.3 Graph expansion 13.4 Expanders from codes 13.5 Ramanujan graphs 13.6 Codes from expanders 13.7 Iterative decoding of graph codes 13.8 Graph codes in concatenated schemes Problems Notes 395 396 401 402 406 409 411 414 420 426 445 14 Trellis and Convolutional Codes 14.1 Labeled directed graphs 14.2 Trellis codes 14.3 Decoding of trellis codes 14.4 Linear finite-state machines 14.5 Convolutional codes 14.6 Encoding of convolutional codes 14.7 Decoding of convolutional codes 14.8 Non-catastrophic generator matrices Problems Notes 452 453 460 466 471 477 479 485 495 501 518 Appendix: Basics inModern Algebra Problems 521 522 Bibliography 527 List of Symbols 553 Index 559 |
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| spelling | Roth, Ron M. ca. 20./21. Jh. Verfasser (DE-588)1256468037 aut Introduction to coding theory Ron M. Roth 1. publ., repr. with corr. Cambridge [u.a.] Cambridge Univ. Press 2007 XI, 566 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Codierungstheorie Coding theory Fehlerkorrekturcode (DE-588)4124917-3 gnd rswk-swf RS-Code (DE-588)4218587-7 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Fehlerkorrekturcode (DE-588)4124917-3 s DE-604 RS-Code (DE-588)4218587-7 s Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016589814&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Roth, Ron M. ca. 20./21. Jh Introduction to coding theory Codierungstheorie Coding theory Fehlerkorrekturcode (DE-588)4124917-3 gnd RS-Code (DE-588)4218587-7 gnd |
| subject_GND | (DE-588)4124917-3 (DE-588)4218587-7 (DE-588)4151278-9 |
| title | Introduction to coding theory |
| title_auth | Introduction to coding theory |
| title_exact_search | Introduction to coding theory |
| title_exact_search_txtP | Introduction to coding theory |
| title_full | Introduction to coding theory Ron M. Roth |
| title_fullStr | Introduction to coding theory Ron M. Roth |
| title_full_unstemmed | Introduction to coding theory Ron M. Roth |
| title_short | Introduction to coding theory |
| title_sort | introduction to coding theory |
| topic | Codierungstheorie Coding theory Fehlerkorrekturcode (DE-588)4124917-3 gnd RS-Code (DE-588)4218587-7 gnd |
| topic_facet | Codierungstheorie Coding theory Fehlerkorrekturcode RS-Code Einführung |
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