The Mathematical Theory of Coding

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.

Preface

Acknowledgments

1. Finite Fields and Coding Theory

1.1 Introduction

1.2 Fields, Extensions, and Polynomials

1.3 Fundamental Properties of Finite Fields

1.4 Vector Spaces Over Finite Fields

1.5 Linear Codes

1.6 Polynomials over Finite Fields

1.7 Cyclic Codes

1.8 Linear Transformations of Vector Spaces over Finite Fields

1.9 Code Invariance under Permutation Groups

1.10 The Polynomial Approach to Coding

1.11 Bounds on Code Dictionaries

1.12 Comments

Exercises

2. Combinatorial Constructions and Coding

2.1 Introduction

2.2 Finite Geometries: Their Collineation Groups and Codes

2.3 Balanced Incomplete Block Designs and Codes

2.4 Latin Squares and Steiner Triple Systems

2.5 Quadratic Residues and Codes

2.6 Hadamard Matrices, Difference Sets, and Their Codes

2.7 Self-Dual and Quasicyclic Codes

2.8 Perfect Codes

2.9 Comments

Exercises

3. Coding and Combinatorics

3.1 Introduction

3.2 General t Designs

3.3 Matroids

3.4 Chains and Chain Groups

3.5 Dual Chain Groups

3.6 Matroids, Graphs, and Coding

3.7 Perfect Codes and t Designs

3.8 Nearly Perfect Codes and t Designs

3.9 Balanced Codes and t Designs

3.10 Equidistant Codes

3.11 Comments

Exercises

4. The Structure of Semisimple Rings

4.1 Introduction

4.2 Rings, Ideals, and the Minimum Condition

4.3 Nilpotent Ideals and the Radical

4.4 The Structure of Semisimple Rings

4.5 The Structure of Simple Rings

4.6 The Group Algebra and Group Characters

4.7 The Structure of Cyclic Codes

4.8 Abelian Codes

4.9 Comments

Exercises

5. Group Representations

5.1 Introduction

5.2 Representation of Groups

5.3 Group Characters

5.4 Orthogonality Relationships and Properties of Group Characters

5.5 Subduced and Induced Representations

5.6 Direct and Semidirect Products

5.7 Real Representations

5.8 Modules, Group Algebras, and Representations

5.9 Comments

Exercises

6. Group Codes for the Gaussian Channel

6.1 Introduction

6.2 Codes for the Gaussian Channel

6.3 Group Codes for the Gaussian Channel

6.4 The Configuration Matrix

6.5 Distance Properties of Group Codes

6.6 The Initial Vector Problem

6.7 Comments

Exercises

Appendix A. The Mobius Inversion Formula

Appendix B. Lucas's Theorem

Appendix C. The Mathieu Groups

References

Index