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| Preface | |
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| Introduction | |
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| Information Theory | |
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| Concepts of Uncertainty and Information | |
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| Concept of uncertainty | |
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| Concept of information | |
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| Formalization of the Concept of Entropy | |
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| Entropy and Information | |
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| Definitions | |
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| Properties of entropy | |
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| Transmission Channels | |
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| Definitions | |
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| Mutual information | |
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| Capacity of a channel | |
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| Transmission over a Noiseless Channel | |
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| Definitions | |
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| Source coding | |
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| Huffman's algorithm | |
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| Transmission over Noisy Channels | |
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| Some definitions and results | |
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| Shannon's Second Theorem | |
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| Application to error correcting codes | |
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| Continuous Channels | |
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| Properties of the entropy function | |
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| Definition of a TDCA channel | |
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| Noise in continuous channels | |
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| Elements of Algebra | |
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| Basic Structures | |
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| Structures, substructures, morphisms | |
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| Cosets, quotient structures and morphism factorization | |
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| Useful Properties for Studying Error Correcting Codes | |
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| The ring Z | |
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| Vector space | |
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| The ring K[X] | |
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| The ring K[h] | |
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| The algebra K[X,Y] | |
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| Group Characters | |
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| Hadamard Matrices | |
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| Krawtchouk Polynomials | |
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| Finite Fields | |
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| Every Finite Field is Commutative | |
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| Prime Field | |
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| Existence of a Finite Field of Cardinal p[superscript n] (p prime, n [actual symbol not reproducible] N) | |
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| Uniqueness of the Finite Field of Cardinality p[superscript n] | |
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| Automorphisms of GF(p[superscript n]) | |
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| Subgroups of GF(p[superscript n]) | |
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| Trace and Norm | |
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| Bases of GF(p[superscript n]) Considered as a Vector Space over GF(p) | |
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| Squares and Non-Squares in GF(p[superscript n]) | |
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| Factorization of Polynomials | |
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| Construction of a non-invertible element of algebra [GF(q)][superscript t] | |
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| Application to polynomial factorization | |
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| Solution of X[superscript 2] + Y[superscript 2] = [negative]k in GF(q) | |
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| Electronics and Computation in GF(q) | |
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| Finite fields and error correcting codes | |
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| Electronics and error correcting codes | |
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| Integration and VLSI: ASICs | |
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| Encoding algorithms and logic schemes | |
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| Encoding | |
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| Classical Error Correcting Codes | |
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| Introduction | |
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| Linear Codes | |
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| Definition and basic properties | |
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| Generator matrix, control matrix | |
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| Projective linear codes | |
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| Extended codes, truncated codes | |
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| Linear codes in systematic form | |
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| Decoding linear codes | |
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| Weight enumerator polynomials and MacWilliams identities | |
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| Regular codes | |
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| Hamming codes | |
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| The binary Golay code | |
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| Reed-Muller codes | |
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| Cyclic Codes | |
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| Polynomial representation of cyclic codes | |
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| Parity-check matrix of a cyclic code | |
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| Encoding using shift registers | |
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| BCH Codes | |
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| Decoding algorithm for BCH codes | |
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| 2-correcting BCH codes | |
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| Reed-Solomon Codes | |
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| Encoding RS codes | |
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| Parity-deck matrix | |
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| Expression of RS codes over GF(p[superscript m]) in GF(p) | |
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| Other Codes | |
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| Fire codes | |
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| Quadratic residue codes | |
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| Shortened RM codes | |
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| Generalized Reed-Solomon codes | |
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| Goppa codes | |
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| Further Algebra | |
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| Multivariable Codes | |
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| Elementary properties | |
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| Semi-simple codes | |
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| Abelian semi-simple codes | |
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| Modular codes | |
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| Modular abelian codes | |
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| Semi-simple codes and Kasami's construction | |
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| Codes in one or more variables | |
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| Grobner Bases | |
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| Definitions and properties | |
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| A characteristic property of Grobner bases | |
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| Reduced Grobner bases | |
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| Existence and uniqueness of the RGB of an ideal | |
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| Application to codes | |
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| Other Error Correcting Codes | |
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| General Results for Multivariable Codes | |
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| Systems of generators: semi-simple codes | |
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| Systems of generators: modular codes | |
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| Generator and parity-check matrices | |
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| Calculation of the dimension of a code | |
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| Minimum distance | |
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| Encoding and coder | |
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| Decoding | |
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| Some Codes | |
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| Product codes | |
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| Fire codes | |
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| The codes of Blaum, Farrell et al. | |
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| Generalization of Kasami's construction | |
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| Concatenated codes | |
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| SDMC codes | |
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| Generalized Beenker codes | |
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| Principal self dual nilpotent codes | |
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| Extended cyclic codes: BCH, RS, GRM | |
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| Sakata primary codes | |
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| Codes over Rings | |
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| Algebraic Geometry and Codes | |
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| Introduction | |
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| Work on the Weil limit | |
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| Construction of new efficient codes | |
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| Projective curves and geometric codes | |
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| BCH codes as geometric codes | |
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| Codes and Combinatorics | |
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| Bounds on the Information Rate of a Code | |
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| Motivation and general issues | |
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| Bounds on the minimum distance | |
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| Combinatorically Regular Structures | |
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| General issues | |
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| t-designs and codes | |
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| Projective planes and codes | |
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| Association schemes | |
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| Distance-regular graphs | |
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| s-Sum sets | |
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| Relations between Combinatorial Structures | |
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| Basic relations | |
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| Generalizations of relations | |
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| Application of Codes | |
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| Coding Systems | |
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| Channels and noise | |
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| Transmitted and received signals | |
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| Protection algorithm | |
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| Error correcting codes | |
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| Maximum likelihood decoding | |
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| Soft decoding: Chasing | |
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| Checklist | |
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| Evaluation Criteria for the Proposed System | |
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| Theoretical curves | |
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| Experimental curves | |
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| Logical diagrams for coder and decoder | |
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| Some Implementations | |
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| The SPOT4 Satellite | |
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| Introduction | |
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| Reasons for using a corrective code | |
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| Coding specification | |
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| Codes selected | |
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| Implementation of the Hamming code | |
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| Optical Communications | |
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| Introduction | |
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| Presentation of modulation | |
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| Application of error correcting codes | |
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| Tables and Curves | |
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| Tables for Finite Fields | |
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| Tables for the fields (GF(2[superscript m]), m = 1 to 12) | |
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| Zech logarithmic tables | |
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| Traces of elements | |
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| Table of Irreducible Polynomials | |
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| Classification by exponent | |
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| Classification by degree | |
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| Error Probability (Gaussian White Noise) | |
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| Performance over a Gaussian Channel | |
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| Performance curves: BCH and RS codes | |
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| Discretization of curves | |
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| Bibliography | |
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| Index | |