E-Book, Englisch, 242 Seiten, Web PDF
Blake / Mullin An Introduction to Algebraic and Combinatorial Coding Theory
1. Auflage 2014
ISBN: 978-1-4832-6029-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark
E-Book, Englisch, 242 Seiten, Web PDF
ISBN: 978-1-4832-6029-7
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark
An Introduction to Algebraic and Combinatorial Coding Theory focuses on the principles, operations, and approaches involved in the combinatorial coding theory, including linear transformations, chain groups, vector spaces, and combinatorial constructions. The publication first offers information on finite fields and coding theory and combinatorial constructions and coding. Discussions focus on quadratic residues and codes, self-dual and quasicyclic codes, balanced incomplete block designs and codes, polynomial approach to coding, and linear transformations of vector spaces over finite fields. The text then examines coding and combinatorics, including chains and chain groups, equidistant codes, matroids, graphs, and coding, matroids, and dual chain groups. The manuscript also ponders on Möbius inversion formula, Lucas's theorem, and Mathieu groups. The publication is a valuable source of information for mathematicians and researchers interested in the combinatorial coding theory.
Autoren/Hrsg.
Weitere Infos & Material
1;Front Cover;1
2;An Introduction to Algebraic and Combinatorial Coding Theory;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;8
6;PREFACE TO THE ORIGINAL EDITION;10
7;ACKNOWLEDGMENTS;12
8;Chapter 1. Finite Fields and Coding Theory;16
8.1;1.1 Introduction;16
8.2;1.2 Fields, Extensions, and Polynomials;17
8.3;1.3 Fundamental Properties of Finite Fields;22
8.4;1.4 Vector Spaces over Finite Fields;25
8.5;1.5 Linear Codes;30
8.6;1.6 Polynomials over Finite Fields;44
8.7;1.7 Cyclic Codes;55
8.8;1.8 Linear Transformations of Vector Spaces over Finite Fields;73
8.9;1.9 Code Invariance under Permutation Groups;78
8.10;1.10 The Polynomial Approach to Coding;81
8.11;1.11 Bounds on Code Dictionaries;98
8.12;1.12 Comments;104
8.13;Exercises;105
9;Chapter 2. Combinatorial Constructions and Coding;110
9.1;2.1 Introduction;110
9.2;2.2 Finite Geometries: Their Collineation Groups and Codes;110
9.3;2.3 Balanced Incomplete Block Designs and Codes;134
9.4;2.4 Latin Squares and Steiner Triple Systems;147
9.5;2.5 Quadratic Residues and Codes;154
9.6;2.6 Hadamard Matrices, Difference Sets, and Their Codes;159
9.7;2.7 Self-Dual and Quasicyclic Codes;167
9.8;2.8 Perfect Codes;176
9.9;2.9 Comments;180
9.10;Exercises;181
10;Chapter 3. Coding and Combinatorics;185
10.1;3.1 Introduction;185
10.2;3.2 General t Designs;185
10.3;3.3 Matroids;188
10.4;3.4 Chains and Chain Groups;190
10.5;3.5 Dual Chain Groups;193
10.6;3.6 Matroids, Graphs, and Coding;194
10.7;3.7 Perfect Codes and t Designs;198
10.8;3.8 Nearly Perfect Codes and t Designs;201
10.9;3.9 Balanced Codes and / Designs;204
10.10;3.10 Equidistant Codes;207
10.11;3.11 Comments;216
10.12;Exercises;216
11;Appendix A: The Möbius Inversion Formula;220
12;Appendix B: Lucas's Theorem;221
13;Appendix C: The Mathieu Groups;222
14;References;230
15;Index;238
