Intro -- Preface -- Acknowledgements -- Contents -- Acronyms -- Part I Theoretical Performance of Error-Correcting Codes -- 1 Bounds on Error-Correction Coding Performance -- 1.1 Gallager's Coding Theorem -- 1.1.1 Linear Codes with a Binomial Weight Distribution -- 1.1.2 Covering Radius of Codes -- 1.1.3 Usefulness of Bounds -- 1.2 Bounds on the Construction of Error-Correcting Codes -- 1.2.1 Upper Bounds -- 1.2.2 Lower Bounds -- 1.2.3 Lower Bounds from Code Tables -- 1.3 Summary -- References -- 2 Soft and Hard Decision Decoding Performance -- 2.1 Introduction -- 2.2 Hard Decision Performance -- 2.2.1 Complete and Bounded Distance Decoding -- 2.2.2 The Performance of Codes on the Binary Symmetric Channel -- 2.3 Soft Decision Performance -- 2.3.1 Performance Assuming a Binomial Weight Distribution -- 2.3.2 Performance of Self-dual Codes -- 2.4 Summary -- References -- 3 Soft Decision and Quantised Soft Decision Decoding -- 3.1 Introduction -- 3.2 Soft Decision Bounds -- 3.3 Examples -- 3.4 A Hard Decision Dorsch Decoder and BCH Codes -- 3.5 Summary -- References -- Part II Code Construction -- 4 Cyclotomic Cosets, the Mattson--Solomon Polynomial, Idempotents and Cyclic Codes -- 4.1 Introduction -- 4.2 Cyclotomic Cosets -- 4.3 The Mattson--Solomon Polynomial -- 4.4 Binary Cyclic Codes Derived from Idempotents -- 4.4.1 Non-Primitive Cyclic Codes Derived from Idempotents -- 4.5 Binary Cyclic Codes of Odd Lengths from 129 to 189 -- 4.6 Summary -- References -- 5 Good Binary Linear Codes -- 5.1 Introduction -- 5.2 Algorithms to Compute the Minimum Hamming Distance of Binary Linear Codes -- 5.2.1 The First Approach to Minimum Distance Evaluation -- 5.2.2 Brouwer's Algorithm for Linear Codes -- 5.2.3 Zimmermann's Algorithm for Linear Codes and Some Improvements -- 5.2.4 Chen's Algorithm for Cyclic Codes -- 5.2.5 Codeword Enumeration Algorithm.

5.3 Binary Cyclic Codes of Lengths 129 len le 189 -- 5.4 Some New Binary Cyclic Codes Having Large Minimum Distance -- 5.5 Constructing New Codes from Existing Ones -- 5.5.1 New Binary Codes from Cyclic Codes of Length 151 -- 5.5.2 New Binary Codes from Cyclic Codes of Length ge 199 -- 5.6 Concluding Observations on Producing New Binary Codes -- 5.7 Summary -- References -- 6 Lagrange Codes -- 6.1 Introduction -- 6.2 Lagrange Interpolation -- 6.3 Lagrange Error-Correcting Codes -- 6.4 Error-Correcting Codes Derived from the Lagrange Coefficients -- 6.5 Goppa Codes -- 6.6 BCH Codes as Goppa Codes -- 6.7 Extended BCH Codes as Goppa Codes -- 6.8 Binary Codes from MDS Codes -- 6.9 Summary -- References -- 7 Reed--Solomon Codes and Binary Transmission -- 7.1 Introduction -- 7.2 Reed--Solomon Codes Used with Binary Transmission-Hard Decisions -- 7.3 Reed--Solomon Codes and Binary Transmission Using Soft Decisions -- 7.4 Summary -- References -- 8 Algebraic Geometry Codes -- 8.1 Introduction -- 8.2 Motivation for Studying AG Codes -- 8.2.1 Bounds Relevant to Algebraic Geometry Codes -- 8.3 Curves and Planes -- 8.3.1 Important Theorems and Concepts -- 8.3.2 Construction of AG Codes -- 8.4 Generalised AG Codes -- 8.4.1 Concept of Places of Higher Degree -- 8.4.2 Generalised Construction -- 8.5 Summary -- References -- 9 Algebraic Quasi Cyclic Codes -- 9.1 Introduction -- 9.2 Background and Notation -- 9.2.1 Description of Double-Circulant Codes -- 9.3 Good Double-Circulant Codes -- 9.3.1 Circulants Based Upon Prime Numbers Congruent to pm3 Modulo 8 -- 9.3.2 Circulants Based Upon Prime Numbers Congruent to +1 mod 8, or -1 mod 8: Cyclic Codes -- 9.4 Code Construction -- 9.4.1 Double-Circulant Codes from Extended Quadratic Residue Codes -- 9.4.2 Pure Double-Circulant Codes for Primes +3 mod 8, or -3 mod 8 -- 9.4.3 Quadratic Double-Circulant Codes.

9.5 Evaluation of the Number of Codewords of Given Weight ƒ -- 9.6 Weight Distributions -- 9.6.1 The Number of Codewords of a Given Weight in Quadratic Double-Circulant Codes -- 9.6.2 The Number of Codewords of a Given Weight in Extended Quadratic Residue Codes -- 9.7 Minimum Distance Evaluation: A Probabilistic Approach -- 9.8 Conclusions -- 9.9 Summary -- References -- 10 Historical Convolutional Codes as Tail-Biting Block Codes -- 10.1 Introduction -- 10.2 Convolutional Codes and Circulant Block Codes -- 10.3 Summary -- References -- 11 Analogue BCH Codes and Direct Reduced Echelon Parity Check Matrix Construction -- 11.1 Introduction -- 11.2 Analogue BCH Codes and DFT Codes -- 11.3 Error-Correction of Bandlimited Data -- 11.4 Analogue BCH Codes Based on Arbitrary Field Elements -- 11.5 Examples -- 11.5.1 Example of Simple (5,3,3) Analogue Code -- 11.5.2 Example of Erasures Correction Using (15,10,4) Binary BCH code -- 11.5.3 Example of (128, 112, 17) Analogue BCH Code and Error-Correction of Audio Data (Music) Subjected to Impulsive Noise -- 11.6 Conclusions and Future Research -- 11.7 Summary -- References -- 12 LDPC Codes -- 12.1 Background and Notation -- 12.1.1 Random Constructions -- 12.1.2 Algebraic Constructions -- 12.1.3 Non-binary Constructions -- 12.2 Algebraic LDPC Codes -- 12.2.1 Mattson--Solomon Domain Construction of Binary Cyclic LDPC Codes -- 12.2.2 Non-Binary Extension of the Cyclotomic Coset-Based LDPC Codes -- 12.3 Irregular LDPC Codes from Progressive Edge-Growth Construction -- 12.4 Quasi-cyclic LDPC Codes and Protographs -- 12.4.1 Quasi-cyclic LDPC Codes -- 12.4.2 Construction of Quasi-cyclic Codes Using a Protograph -- 12.5 Summary -- References -- Part III Analysis and Decoders -- 13 An Exhaustive Tree Search for Stopping Sets of LDPC Codes -- 13.1 Introduction and Preliminaries.

13.2 An Efficient Tree Search Algorithm -- 13.2.1 An Efficient Lower Bound -- 13.2.2 Best Next Coordinate Position Selection -- 13.3 Results -- 13.3.1 WiMax LDPC Codes -- 13.4 Conclusions -- 13.5 Summary -- References -- 14 Erasures and Error-Correcting Codes -- 14.1 Introduction -- 14.2 Derivation of the PDF of Correctable Erasures -- 14.2.1 Background and Definitions -- 14.2.2 The Correspondence Between Uncorrectable Erasure Patterns and Low-Weight Codewords -- 14.3 Probability of Decoder Error -- 14.4 Codes Whose Weight Enumerator Coefficients Are Approximately Binomial -- 14.5 MDS Shortfall for Examples of Algebraic, LDPC and Turbo Codes -- 14.5.1 Turbo Codes with Dithered Relative Prime (DRP) Interleavers -- 14.5.2 Effects of Weight Spectral Components -- 14.6 Determination of the dmin of Any Linear Code -- 14.7 Summary -- References -- 15 The Modified Dorsch Decoder -- 15.1 Introduction -- 15.2 The Incremental Correlation Dorsch Decoder -- 15.3 Number of Codewords that Need to Be Evaluated to Achieve ƒ -- 15.4 Results for Some Powerful Binary Codes -- 15.4.1 The (136, 68, 24) Double-Circulant Code -- 15.4.2 The (255, 175, 17) Euclidean Geometry (EG) Code -- 15.4.3 The (513, 467, 12) Extended Binary Goppa Code -- 15.4.4 The (1023, 983, 9) BCH Code -- 15.5 Extension to Non-binary Codes -- 15.5.1 Results for the (63, 36, 13) GF(4) BCH Code -- 15.6 Conclusions -- 15.7 Summary -- References -- 16 A Concatenated Error-Correction System Using the 69640972 u69640972 u+v69640972 Code Construction -- 16.1 Introduction -- 16.2 Description of the System -- 16.3 Concatenated Coding and Modulation Formats -- 16.4 Summary -- References -- Part IV Applications -- 17 Combined Error Detection and Error-Correction -- 17.1 Analysis of Undetected Error Probability -- 17.2 Incremental-Redundancy Coding System -- 17.2.1 Description of the System -- 17.3 Summary.

References -- 18 Password Correction and Confidential Information Access System -- 18.1 Introduction and Background -- 18.2 Details of the Password System -- 18.3 Summary -- References -- 19 Variations on the McEliece Public Key Cryptoystem -- 19.1 Introduction and Background -- 19.1.1 Outline of Different Variations of the Encryption System -- 19.2 Details of the Encryption System -- 19.3 Reducing the Public Key Size -- 19.4 Reducing the Cryptogram Length Without Loss of Security -- 19.5 Security of the Cryptosystem -- 19.5.1 Probability of a k timesk Random Matrix Being Full Rank -- 19.5.2 Practical Attack Algorithms -- 19.6 Applications -- 19.7 Summary -- References -- 20 Error-Correcting Codes and Dirty Paper Coding -- 20.1 Introduction and Background -- 20.2 Description of the System -- 20.3 Summary -- References -- Index.

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