NON-BINARY ERROR CONTROL CODING FOR WIRELESS COMMUNICATION AND DATA STORAGE -- Contents -- Biographies -- Preface -- Acknowledgements -- 1 Information, Channel Capacity and Channel Modelling -- 1.1 Introduction -- 1.1.1 Information Theory [1] -- 1.1.2 Definition of Information [1, 2] -- 1.2 Measure of Information [1-3] -- 1.2.1 Average Information -- 1.2.2 The Entropy of a Binary Source -- 1.2.3 Mutual Information -- 1.3 Channel Capacity -- 1.3.1 Binary Symmetric Input Channel -- 1.3.2 Binary Erasure Channel (BEC) -- 1.3.3 The 4-ary Symmetric Channel [1, 5] -- 1.3.4 Binary Input Capacity of a Continuous Channel (Channel Capacity) -- 1.3.5 Channel Capacity in Fading Environments -- 1.4 Channel Modelling -- 1.4.1 Classification of Models -- 1.5 Definition of a Communications Channel and its Parameters -- 1.5.1 Doppler Shift -- 1.5.2 Scattering -- 1.5.3 Angle of Arrival and Spectra -- 1.5.4 Multipath Channel and Tapped Delay Line Model -- 1.5.5 Delay Spread -- 1.5.6 The Fixed Broadband Wireless Access Channel -- 1.5.7 UMTS Mobile Wireless Channel Model -- 1.5.8 Simulating the Fixed Broadband Wireless Access Channel Model -- 1.5.9 Simulating the UMTS Mobile Radio Channel -- 1.6 (Multiple-Input-Multiple-Output) (MIMO) Channel -- 1.7 Magnetic Storage Channel Modelling -- 1.7.1 Longitudinal Recording -- 1.7.2 (Partial Response Maximum Likelihood) (PRML) Detection -- 1.8 Conclusions -- References -- 2 Basic Principles of Non-Binary Codes -- 2.1 Introduction to Algebraic Concepts -- 2.1.1 Groups -- 2.1.2 Rings -- 2.1.3 Ring of Polynomials -- 2.1.4 Fields -- 2.1.5 Primitive Polynomials -- 2.1.6 Minimal Polynomials and Cyclotomic Cosets -- 2.1.7 Subfields -- 2.2 Algebraic Geometry Principles -- 2.2.1 Projective and Affine Space -- 2.2.2 Projective and Affine Curves -- 2.2.3 Finding Points on an Affine Curve -- 2.2.4 Rational Functions on Curves.

2.2.5 Riemann-Roch Theorem -- 2.2.6 The Zero Order of a Monomial -- 2.2.7 AG Code Parameters -- 2.3 Conclusions -- References -- 3 Non-Binary Block Codes -- 3.1 Introduction -- 3.2 Fundamentals of Block Codes -- 3.2.1 Generator and Parity Check Matrices -- 3.2.2 Decoding Block Codes -- 3.3 Cyclic Codes -- 3.3.1 Polynomials -- 3.3.2 Systematic Cyclic Codes -- 3.4 Bose-Chaudhuri-Hocquenghem (BCH) Codes -- 3.5 Reed-Solomon Codes -- 3.6 Decoding Reed-Solomon Codes -- 3.6.1 Euclid's Algorithm -- 3.6.2 Berlekamp-Massey's Algorithm -- 3.6.3 Determining the Error Magnitudes Using Forney's Formula -- 3.7 Coded Modulation -- 3.7.1 Block Coded Modulation (BCM) Codes -- 3.7.2 Multi-Level Block Coding -- 3.7.3 Set Partitioning -- 3.7.4 Construction of a Block Code Trellis -- 3.7.5 Non-Binary BCM Codes -- 3.8 Conclusions -- References -- 4 Algebraic-Geometric Codes -- 4.1 Introduction -- 4.2 Construction of Algebraic-Geometric Codes -- 4.2.1 Simplified Construction of AG Codes -- 4.2.2 Comparison of AG Codes with Reed-Solomon Codes -- 4.2.3 Systematic Algebraic-Geometric Codes -- 4.3 Decoding Algebraic-Geometric Codes -- 4.3.1 Modification of the Sets F and G -- 4.4 Majority Voting -- 4.5 Calculating the Error Magnitudes -- 4.6 Complete Hard-Decision Decoding Algorithm for Hermitian Codes -- 4.7 Simulation Results -- 4.8 Conclusions -- References -- 5 List Decoding -- 5.1 Introduction -- 5.2 List Decoding of Reed-Solomon Codes Using the Guruswami-Sudan Algorithm -- 5.2.1 Weighted Degrees and Lexicographic Orders -- 5.2.2 Interpolation Theorem -- 5.2.3 Iterative Polynomial Construction -- 5.2.4 Complexity Reduced Modification -- 5.2.5 Factorization -- 5.2.6 Recursive Coefficient Search -- 5.3 Soft-Decision List Decoding of Reed-Solomon Codes Using the K ̈otter-Vardy Algorithm -- 5.3.1 Mapping Reliability Values into Multiplicity Values.

5.3.2 Solution Analysis for Soft-Decision List Decoding -- 5.3.3 Simulation Results -- 5.4 List Decoding of Algebraic-Geometric Codes -- 5.5 Determining the Corresponding Coefficients -- 5.6 Complexity Reduction Interpolation -- 5.7 General Factorization -- 5.8 Soft-Decision List Decoding of Hermitian Codes -- 5.8.1 System Solution -- 5.8.2 Simulation Results -- 5.9 Conclusions -- References -- 6 Non-Binary Low-Density Parity Check Codes -- 6.1 Introduction -- 6.2 Construction of Binary LDPC Codes - Random and Structured Methods -- 6.2.1 Tanner Graph Representation -- 6.2.2 Structured Construction Methods -- 6.2.3 LDPC Codes from Class 1 Bose BIBD -- 6.2.4 Constructing the Generator Matrix of a Binary LDPC Code -- 6.3 Decoding of Binary LDPC Codes Using the Belief Propagation Algorithm -- 6.3.1 Initialization of the Belief Propagation Algorithm -- 6.3.2 The Horizontal Step -- 6.3.3 The Vertical Step -- 6.3.4 Reducing the Decoding Complexity Using Fast Fourier Transforms -- 6.4 Construction of Non-Binary LDPC Codes Defined Over Finite Fields -- 6.4.1 Construction of Non-Binary LDPC Codes from Reed-Solomon Codes -- 6.5 Decoding Non-Binary LDPC Codes with the Sum-Product Algorithm -- 6.5.1 Received Symbol Likelihoods -- 6.5.2 Permutation of Likelihoods -- 6.5.3 Factor Graph of a Non-Binary LDPC Code -- 6.5.4 The Fast Fourier Transform for the Decoding of Non-Binary LDPC Codes -- 6.6 Conclusions -- References -- 7 Non-Binary Convolutional Codes -- 7.1 Introduction -- 7.1.1 The Viterbi Algorithm -- 7.1.2 Trellis Coded Modulation -- 7.1.3 TCM Based on Rings of Integers -- 7.1.4 Ring-TCM Codes for M-PSK -- 7.1.5 Ring-TCM Codes Using Quadrature Amplitude Modulation -- 7.1.6 Searching for Good Ring-TCM Codes -- 7.1.7 Genetic Algorithm -- 7.1.8 Performance of Ring-TCM Codes on Urban Fading Channels -- 7.2 Space-Time Coding Modulation -- 7.2.1 Introduction.

7.2.2 Space-Time Codes Model -- 7.2.3 Performance of ST-RTCM Codes Using QPSK for MIMO Urban Environments -- 7.2.4 Ideal CSI and Slow Fading -- 7.2.5 Log-Likelihood Function -- 7.2.6 Maximum-Likelihood Sequence Detection -- 7.2.7 ST Codes Pairwise Error Probability -- 7.2.8 ST Code Design Criteria for Slow Fading Channels -- 7.2.9 ST Code Design Criteria for Fast Fading Channels -- 7.2.10 Space-Time Ring-TCM Codes for QAM in Fading Channels -- 7.3 Conclusions -- References -- 8 Non-Binary Turbo Codes -- 8.1 Introduction -- 8.2 The Turbo Encoder -- 8.2.1 Notation -- 8.3 The Turbo Decoder -- 8.3.1 Log-Likelihood Ratios (LLRs) -- 8.3.2 Maximum A Posteriori (MAP) Decoding -- 8.3.3 Max-Log MAP -- 8.3.4 Log MAP Algorithm -- 8.3.5 Iterative Log MAP Decoding -- 8.4 Non-Binary Turbo Codes -- 8.4.1 Multi-Dimensional Log-Likelihood Ratios -- 8.4.2 Non-Binary Iterative Turbo Decoding -- 8.5 Conclusions -- References -- Index.