Coding Theory: Algorithms, Architectures and Applications

TLDR: The aim of this presentation is to clarify the role of encoding in the development of knowledge representation and to provide some examples of how information theory can be used to improve the quality of coding.

Abstract: Preface. 1 Introduction. 1.1 Communication Systems. 1.2 Information Theory. 1.2.1 Entropy. 1.2.2 Channel Capacity. 1.2.3 Binary Symmetric Channel. 1.2.4 AWGN Channel. 1.3 A Simple Channel Code. 2 Algebraic Coding Theory. 2.1 Fundamentals of Block Codes. 2.1.1 Code Parameters. 2.1.2 Maximum Likelihood Decoding. 2.1.3 Binary Symmetric Channel. 2.1.4 Error Detection and Error Correction. 2.2 Linear Block Codes. 2.2.1 Definition of Linear Block Codes. 2.2.2 Generator Matrix. 2.2.3 Parity Check Matrix. 2.2.4 Syndrome and Cosets. 2.2.5 Dual Code. 2.2.6 Bounds for Linear Block Codes. 2.2.7 Code Constructions. 2.2.8 Examples of Linear Block Codes. 2.3 Cyclic Codes. 2.3.1 Definition of Cyclic Codes. 2.3.2 Generator Polynomial. 2.3.3 Parity Check Polynomial. 2.3.4 Dual Codes. 2.3.5 Linear Feedback Shift Registers. 2.3.6 BCH Codes. 2.3.7 Reed-Solomon Codes. 2.3.8 Algebraic Decoding Algorithm. 2.4 Summary. 3 Convolutional Codes. 3.1 Encoding of Convolutional Codes. 3.1.1 Convolutional Encoder. 3.1.2 Generator Matrix in Time-Domain. 3.1.3 State Diagram of a Convolutional Encoder. 3.1.4 Code Termination. 3.1.5 Puncturing. 3.1.6 Generator Matrix in D -Domain. 3.1.7 Encoder Properties. 3.2 Trellis Diagram and Viterbi's Algorithm. 3.2.1 Minimum Distance Decoding. 3.2.2 Trellises. 3.2.3 Viterbi Algorithm. 3.3 Distance Properties and Error Bounds. 3.3.1 Free Distance. 3.3.2 Active Distances. 3.3.3 Weight Enumerators for Terminated Codes. 3.3.4 Path Enumerators. 3.3.5 Pairwise Error Probability. 3.3.6 Viterbi Bound. 3.4 Soft Input Decoding. 3.4.1 Euclidean Metric. 3.4.2 Support of Punctured Codes. 3.4.3 Implementation Issues. 3.5 Soft Output Decoding. 3.5.1 Derivation of APP Decoding. 3.5.2 APP Decoding in the Log-Domain. 3.6 Convolutional Coding in Mobile Communications. 3.6.1 Coding of Speech Data. 3.6.2 Hybrid ARQ. 3.6.3 EGPRS Modulation and Coding. 3.6.4 Retransmission Mechanism. 3.6.5 Link Adaptation. 3.6.6 Incremental Redundancy. 3.7 Summary. 4 Turbo Codes. 4.1 LDPC Codes. 4.1.1 Codes Based on Sparse Graphs. 4.1.2 Decoding for the Binary Erasure Channel. 4.1.3 Log-Likelihood Algebra. 4.1.4 Belief Propagation. 4.2 A First Encounter with Code Concatenation. 4.2.1 Product Codes. 4.2.2 Iterative Decoding of Product Codes. 4.3 Concatenated Convolutional Codes. 4.3.1 Parallel Concatenation. 4.3.2 The UMTS Turbo Code. 4.3.3 Serial Concatenation. 4.3.4 Partial Concatenation. 4.3.5 Turbo Decoding. 4.4 EXIT Charts. 4.4.1 Calculating an EXIT Chart. 4.4.2 Interpretation. 4.5 Weight Distribution. 4.5.1 Partial Weights. 4.5.2 ExpectedWeight Distribution. 4.6 Woven Convolutional Codes. 4.6.1 Encoding Schemes. 4.6.2 Distance Properties of Woven Codes. 4.6.3 Woven Turbo Codes. 4.6.4 Interleaver Design. 4.7 Summary. 5 Space-Time Codes. 5.1 Introduction. 5.1.1 Digital Modulation Schemes. 5.1.2 Diversity. 5.2 Spatial Channels. 5.2.1 Basic Description. 5.2.2 Spatial Channel Models. 5.2.3 Channel Estimation. 5.3 Performance Measures. 5.3.1 Channel Capacity. 5.3.2 Outage Probability and Outage Capacity. 5.3.3 Ergodic Error Probability. 5.4 Orthogonal Space-Time Block Codes. 5.4.1 Alamouti's Scheme. 5.4.2 Extension to more than two Transmit Antennas. 5.4.3 Simulation Results. 5.5 Spatial Multiplexing. 5.5.1 General Concept. 5.5.2 Iterative APP Preprocessing and Per-Layer Decoding. 5.5.3 Linear Multi-Layer Detection. 5.5.4 Original Bell Labs Layered Space Time (BLAST) Detection. 5.5.5 QL Decomposition and Interference Cancellation. 5.5.6 Performance of Multi-Layer Detection Schemes. 5.5.7 Unified Description by Linear Dispersion Codes. 5.6 Summary. A. Algebraic Structures. A.1 Groups, Rings and Finite Fields. A.1.1 Groups. A.1.2 Rings. A.1.3 Finite Fields. A.2 Vector Spaces. A.3 Polynomials and Extension Fields. A.4 Discrete Fourier Transform. B. Linear Algebra. C. Acronyms. Bibliography . Index.