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.

Preface

Preface to the Original Edition


Acknowledgment


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


Appendix A. The Möbius Inversion Formula


Appendix B. Lucas's Theorem


Appendix C. The Mathieu Groups


References


Index