1 Introduction and Motivation 1.1 Definition,Related Fields 1.2 Basic Principles 1.2.1 Schemes 1.2.2 How to Add Redundancy 1.2.3 Applications 1.3 Historical Notes2 Fundamentals of Block Coding 2.1 General Assumptions 2.2 Transmission Channels 2.2.1 Discrete-Time AWGN Channel 2.2.2 Binary Symmetric Channel (BSC) 2.2.3 Channels with Memory 2.3 Motivation for Coding 2.4 Fundamentals of Block Coding 2.4.1 Code and Encoding 2.4.2 Decoding
3 Introduction to Finite Fields I 3.1 Group 3.1.1 Orders of Elements and Cycles 3.1.2 Subgroups, Cosets 3.2 Field 3.3 Vector Spaces
4 Linear Block Codes 4.1 Generator Matrix 4.2 Distance Properties 4.3 Elementary Operations 4.4 Parity-Check Matrix 4.5 Dual Codes 4.6 Syndrome Decoding 4.7 Error Probability and Coding Gain 4.7.1 Error Detection 4.7.2 Error Correction – BMD 4.7.3 Error Correction – ML Decoding 4.7.4 Coding Gain 4.7.5 Asymptotic Results 4.8 Modifications of Codes 4.9 Bounds on the Minimum Distance 4.10 Examples for Linear Block Codes 4.10.1 Binary Hamming Codes (q=2) 4.10.2 Simplex Codes 4.10.3 Ternary Golay Code 4.10.4 Reed-Muller Codes
5 Linear Cyclic Codes 5.1 Modular Arithmetic 5.2 Generator Polynomial 5.3 Parity-Check Polynomial 5.4 Dual Codes 5.5 Discrete Systems over Fq 5.6 Encoders for Cyclic Codes 5.6.1 Generator Matrix 5.6.2 Non-Systematic Encoding 5.6.3 Systematic Encoding 5.6.4 Systematic Encoding Using h(x) 5.7 Syndrome Decoding 5.7.1 Syndrome 5.7.2 Decoding Strategies 5.8 Examples for Linear Cyclic Block Codes 5.8.1 Repetition Code and Single Parity-Check Code 5.8.2 Binary Hamming Codes 5.8.3 Simplex Codes 5.8.4 Golay Codes 5.8.5 CRC Codes
6 Introduction to Finite Fields II 6.1 Extension Fields 6.2 Polynomials over Finite Fields 6.3 Primitive Element 6.4 Existence of Finite Fields 6.5 Finite Fields Arithmetic 6.6 Minimal Polynomials, Conjugate Elements, and Cyclotomic Cosets 6.7 Summary of Important Properties of Finite Fields 6.8 (Discrete) Fourier Transform over Finite Fields
7 BCH and RS Codes 7.1 The BCH Bound 7.2 Reed-Solomon Codes 7.3 BCH Codes 7.4 Algebraic Decoding of BCH Codes and RS Codes 7.4.1 Basic Idea 7.4.2 The Berlekamp-Massey Algorithm 7.5 Application: Channel Coding for CD and DVD 7.5.1 Error Correction for the CD 7.5.2 Error Correction for the DVD
8 Convolutional Codes 8.1 Discrete Systems over F 8.2 Trellis Coding 8.3 Encoders for Convolutional Codes 8.4 (Optimal) Decoding of Convolutional Codes 8.4.1 Maximum-Likelihood Sequence Estimation (MLSE) 8.4.2 Maximum A-Posteriori Symbol-by-Symbol Estimation
9 Codes with Iterative Decoding 9.1 State of the Art 9.2 Preliminaries 9.2.1 Check Equations 9.2.2 Repetition Code, Parallel Channels 9.2.3 Log-Likelihood Ratios(LLR) 9.3 Turbo Codes 9.4 LDPC Codes