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Error-Correction Coding and Decoding : Bounds, Codes, Decoders, Analysis and Applications.

Bibliographic Details
Main Author: Tomlinson, Martin.
Other Authors: Tjhai, Cen Jung., Ambroze, Marcel A., Ahmed, Mohammed., Jibril, Mubarak.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2017.
Edition:1st ed.
Series:Signals and Communication Technology Series
Subjects:
Electronic books.
Online Access:Click to View
Table of Contents:
  • 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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