Introduction to coding theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boca Raton ; London ; New York
CRC Press
[2017]
|
| Ausgabe: | second edition |
| Schriftenreihe: | Discrete mathematics and its applications
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| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverzeichnis Seite 487-501 |
| Beschreibung: | xxv, 512 Seiten Diagramme |
| ISBN: | 9781482299809 |
Internformat
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Datensatz im Suchindex
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| adam_text | Contents Preface xxiii Acknowledgments xxiv About the author xxvi I An elementary introduction to coding 1 The concept of coding 1.1 1.2 1.3 1.4 1.5 1.6 Bitstrings and binary operations .......................................... The Hamming distance ......................................................... Binary codes ........................................................................... Error-correcting codes in general .......................................... The binary symmetric channel ............................................. The sphere-packing bound...................................................... 2 Binary linear codes 2.1 2.2 2.3 2.4 2.5 2.6 The concept of binary linear codes ....................................... Block coding ........................................................................... The effect of coding ............................................................... Duality .................................................................................... Binary Hamming and Simplex codes .................................... Principle of duality.................................................................. 3 General linear codes 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Prime fields.............................................................................. Finite fields .............................................................................. Linear codes over finite fields ................................................ Duality and orthogonal arrays................................................ Weight
distribution.................................................................. The game of SET..................................................................... Syndrome decoding.................................................................. 1 3 3 7 9 12 14 19 23 23 27 29 30 33 37 41 41 43 47 50 58 63 66 4 Singleton bound and Reed-Solomon codes 71 5 Recursive constructions I 81 5.1 5.2 Shortening and puncturing ................................................... Concatenation ........................................................................ 81 87 ix
x Contents 6 Universal hashing 93 7 Designs and the binary Golay code 97 8 Shannon entropy 101 9 Asymptotic results 113 10 Three-dimensional codes, projective planes 125 11 Summary and outlook 131 11 133 Theory and applications of codes 12 Subfield codes and trace codes 12.1 The trace .................................................................................... 12.2 Trace codes and subfield codes ............................................... 12.3 Galois closed codes.................................................................... 12.4 Automorphism groups .............................................................. 135 135 140 143 146 13 Cyclic codes 13.1 Some primitive cyclic codes of length 15 ............................... 13.2 Theory of cyclic codes .............................................................. 13.3 Decoding BCH codes................................................................. 13.4 Constacyclic codes .................................................................... 13.5 Remarks....................................................................................... 151 151 154 170 182 186 14 Recursive constructions, covering radius 14.1 Construction X........................................................................... 14.2 Covering radius........................................................................... 189 189 198 15 The 15.1 15.2 15.3 15.4 linear programming method Introduction to linear programming ..................................... The Fourier transform
.............................................................. Some explicit LP bounds ........................................................ The bound of four .................................................................... 205 205 221 230 232 16 OA in statistics and computer science 16.1 О A and independent random variables.................................. 16.2 Linear shift register sequences.................................................. 16.3 Cryptography and S boxes........................................................ 16.4 Two-point-based sampling........................................................ 16.5 Resilient functions .................................................................... 16.6 Derandomization of algorithms ............................................... 16.7 Authentication and universal hashing .................................. 239 239 242 250 254 256 265 270
Contents 17 The geometric description of linear codes 17.1 Linear codes as sets of points ................................................ 17.2 Quadratic forms, bilinear forms and caps ........................... 17.3 Caps: Constructions and bounds .......................................... 18 Additive codes and network codes 18.1 Basic constructions and applications .................................... 18.2 The cyclic theory of additive codes....................................... 18.2.1 Code equivalence and cyclicity.................................... 18.2.2 The linear case m = 1 ................................................ 18.3 Additive quaternary codes: The geometric approach .... 18.4 Quantum codes........................................................................ 18.5 Network codes and subspace codes ....................................... III Codes and algebraic curves 19 Introduction 19.1 19.2 Polynomial equations and function fields.............................. Places of the rational function field....................................... 20 Function fields, their places and valuations 20.1 20.2 20.3 20.4 General facts ........................................................................... Divisors and the genus............................................................ The Riemann-Roch theorem................................................... Some hyperelliptic equations ................................................ 21 Determining the genus 21.1 21.2 21.3 21.4 21.5 Algebraic extensions of function fields ................................. The
hyperelliptic case ............................................................ The Kloosterman codes and curves....................................... Subrings and integrality ......................................................... The Riemann-Hurwitz formula ............................................. 22 AG codes, Weierstraß points and universal hashing 22.1 22.2 22.3 22.4 22.5 22.6 The basic construction............................................................ Pole orders .............................................................................. Examples of function fields and projective equations .... The automorphism group ...................................................... AG codes and universal hashing .......................................... The Hasse-Weil bound............................................................ 23 The last chapter 23.1 List decoding ........................................................................... 23.2 Expander codes........................................................................ 23.3 tms-nets.................................................................................... 23.4 Sphere packings........................................................................ xi 285 285 312 332 349 349 360 360 370 373 380 389 395 397 397 401 405 405 409 414 417 421 421 423 424 426 427 431 431 432 433 440 443 444 447 447 450 452 456
Contents xii 23.5 23.6 23.7 23.8 23.9 23.10 Permutation codes .................................................................... Designs ....................................................................................... Nonlinear codes.......................................................................... Some highly symmetric codes.................................................. Small fields ................................................................................. Short codes ................................................................................. 466 468 470 479 482 483 References 487 Index 503
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| spelling | Bierbrauer, Jürgen 1948- (DE-588)143427210 aut Introduction to coding theory Jürgen Bierbrauer, Michigan Technological University, Houghton, USA second edition Boca Raton ; London ; New York CRC Press [2017] xxv, 512 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Discrete mathematics and its applications Literaturverzeichnis Seite 487-501 Coding theory Codierungstheorie (DE-588)4139405-7 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Codierungstheorie (DE-588)4139405-7 s DE-604 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=029251691&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Bierbrauer, Jürgen 1948- Introduction to coding theory Coding theory Codierungstheorie (DE-588)4139405-7 gnd |
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| title | Introduction to coding theory |
| title_auth | Introduction to coding theory |
| title_exact_search | Introduction to coding theory |
| title_full | Introduction to coding theory Jürgen Bierbrauer, Michigan Technological University, Houghton, USA |
| title_fullStr | Introduction to coding theory Jürgen Bierbrauer, Michigan Technological University, Houghton, USA |
| title_full_unstemmed | Introduction to coding theory Jürgen Bierbrauer, Michigan Technological University, Houghton, USA |
| title_short | Introduction to coding theory |
| title_sort | introduction to coding theory |
| topic | Coding theory Codierungstheorie (DE-588)4139405-7 gnd |
| topic_facet | Coding theory Codierungstheorie Einführung |
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