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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 | |