Blake / Mullin The Mathematical Theory of Coding

1. Auflage 2014
ISBN: 978-1-4832-6059-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 368 Seiten, Web PDF

The Mathematical Theory of Coding focuses on the application of algebraic and combinatoric methods to the coding theory, including linear transformations, vector spaces, and combinatorics. The publication first offers information on finite fields and coding theory and combinatorial constructions and coding. Discussions focus on self-dual and quasicyclic codes, quadratic residues and codes, balanced incomplete block designs and codes, bounds on code dictionaries, code invariance under permutation groups, and linear transformations of vector spaces over finite fields. The text then takes a look at coding and combinatorics and the structure of semisimple rings. Topics include structure of cyclic codes and semisimple rings, group algebra and group characters, rings, ideals, and the minimum condition, chains and chain groups, dual chain groups, and matroids, graphs, and coding. The book ponders on group representations and group codes for the Gaussian channel, including distance properties of group codes, initial vector problem, modules, group algebras, andrepresentations, orthogonality relationships and properties of group characters, and representation of groups. The manuscript is a valuable source of data for mathematicians and researchers interested in the mathematical theory of coding.

Autoren/Hrsg.

Blake, Ian F.

Mullin, Ronald C.

Inhaltsverzeichnis

1;Front Cover;1
2;The Mathematical Theory of Coding;4
3;Copyright Page;5
4;Table of Contents;6
5;Dedication;5
6;Preface;10
7;Acknowledgments;12
8;Chapter 1. Finite Fields and Coding Theory;14
8.1;1.1 Introduction;14
8.2;1.2 Fields, Extensions, and Polynomials;15
8.3;1.3 Fundamental Properties of Finite Fields;20
8.4;1.4 Vector Spaces over Finite Fields;23
8.5;1.5 Linear Codes;28
8.6;1.6 Polynomials over Finite Fields;42
8.7;1.7 Cyclic Codes;53
8.8;1.8 Linear Transformations of Vector Spaces over Finite Fields;71
8.9;1.9 Code Invariance under Permutation Groups;76
8.10;1.10 The Polynomial Approach to Coding;79
8.11;1.11 Bounds on Code Dictionaries;96
8.12;1.12 Comments;102
8.13;Exercises;103
9;Chapter 2. Combinatorial Constructions and Coding;108
9.1;2.1 Introduction;108
9.2;2.2 Finite Geometries: Their Collineation Groups and Codes;108
9.3;2.3 Balanced Incomplete Block Designs and Codes;132
9.4;2.4 Latin Squares and Steiner Triple Systems;145
9.5;2.5 Quadratic Residues and Codes;152
9.6;2.6 Hadamard Matrices, Difference Sets, and Their Codes;157
9.7;2.7 Self-Dual and Quasicyclic Codes;165
9.8;2.8 Perfect Codes;174
9.9;2.9 Comments;178
9.10;Exercises;179
10;Chapter 3. Coding and Combinatorics;183
10.1;3.1 Introduction;183
10.2;3.2 General t Designs;183
10.3;3.3 Matroids;186
10.4;3.4 Chains and Chain Groups;188
10.5;3.5 Dual Chain Groups;191
10.6;3.6 Matroids, Graphs, and Coding;192
10.7;3.7 Perfect Codes and t Designs;196
10.8;3.8 Nearly Perfect Codes and t Designs;199
10.9;3.9 Balanced Codes and t Designs;202
10.10;3.10 Equidistant Codes;205
10.11;3.11 Comments;214
10.12;Exercises;214
11;Chapter 4. The Structure of Semisimpie Rings;217
11.1;4.1 Introduction;217
11.2;4.2 Rings, Ideals, and the Minimum Condition;218
11.3;4.3 Nilpotent Ideals and the Radical;220
11.4;4.4 The Structure of Semisimpie Rings;223
11.5;4.5 The Structure of Simple Rings;228
11.6;4.6 The Group Algebra and Group Characters;230
11.7;4.7 The Structure of Cyclic Codes;235
11.8;4.8 Abelian Codes;240
11.9;4.9 Comments;257
11.10;Exercises;257
12;Chapter 5. Group Representations;259
12.1;5.1 Introduction;259
12.2;5.2 Representation of Groups;259
12.3;5.3 Group Characters;266
12.4;5.4 Orthogonality Relationships and Properties of Group Characters;272
12.5;5.5 Subduced and Induced Representations;282
12.6;5.6 Direct and Semidirect Products;286
12.7;5.7 Real Representations;289
12.8;5.8 Modules, Group Algebras, and Representations;297
12.9;5.9 Comments;303
12.10;Exercises;303
13;Chapter 6. Group Codes for the Gaussian Channel;305
13.1;6.1 Introduction;305
13.2;6.2 Codes for the Gaussian Channel;306
13.3;6.3 Group Codes for the Gaussian Channel;310
13.4;6.4 The Configuration Matrix;317
13.5;6.5 Distance Properties of Group Codes;327
13.6;6.6 The Initial Vector Problem;331
13.7;6.7 Comments;339
13.8;Exercises;339
14;APPENDIX A: The Möbius Inversion Formula;342
15;APPENDIX B: Lucas's Theorem;343
16;APPENDIX C: The Mathieu Groups;344
17;References;352
18;Index;362