An Introduction to Algebraic and Combinatorial Coding Theory

PrefacePreface to the Original EditionAcknowledgment1. 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 Exercises2. 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 Exercises3. 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 ExercisesAppendix A. The Möbius Inversion FormulaAppendix B. Lucas's TheoremAppendix C. The Mathieu GroupsReferencesIndex