Historical Background Codes Predating Hamming Codes Leading to ASCII BCD Codes Digital Arithmetic Number Systems Boolean and Bitwise Operations Checksum Ring Counters Residues, Residue Classes, and Congruences Integral Approximations Lexicographic Order Linear Codes Linear Vector Spaces over Finite Fields Communication Channels Some Useful Definitions Linear Codes Vector Operations Sphere Packing Hamming Codes Error Correcting Codes Hamming (7,4) Code Hamming (11,7) Code General Algorithm Hamming''s Original Algorithm Equivalent Codes q-ary Hamming Codes Extended Hamming Codes SEC-DED Codes Hamming (8,4) Code Hamming (13,8) Code Hamming (32,26) Code Hamming (72,64) Code Hsiao Code Product Notes Uses of Hamming Codes Bounds in Coding Theory Definitions Sphere-Packing Bound Johnson Bound Gilbert-Varshamov Bound Hamming Bound Singleton Bound Plotkin Bound Griesmer Bound Zyablov Bound Bounds in F2n Reiger Bound Krawtchouk Polynomials Linear Programming Bound Stochastic Bounds for SEC-DED Codes Golay Codes Perfect Codes Geometrical Representation Other Construction Methods Finite-State Codes MacWilliams'' Identity Golay''s Original Algorithm Structure of Linear Codes Galois Fields Finite Fields Construction of Galois Fields Galois Fields of Order p Prime Fields Binary Fields Arithmetic in Galois Fields Polynomials Polynomial Codes Matrix Codes Matrix Group Codes Encoding and Decoding Matrices Decoding Procedure Hadamard Code Hadamard Transform Hexacode Lexicodes Octacode Simplex Codes Block Codes Cyclic Codes Definition Construction of Cyclic Codes Methods for Describing Cyclic Codes Quadratic-Residue Codes BCH Codes Binary BCH Codes Extended Finite Fields Construction of BCH Codes General Definition General Algorithm Reed-Muller Codes Boolean Polynomials RM Encoding Generating Matrices for RM Codes Properties of RM Codes Classification of RM Codes Decoding of RM Codes Recursive Definition Probability Analysis Burst Errors Reed-Solomon Codes Definition Reed-Solomon''s Original Approach Parity Check Matrix RS Encoding and Decoding Burst Errors Erasures Concatenated Systems Applications Belief Propagation Rational Belief Belief Propagation Stopping Time Probability Density Function Log-Likelihood Ratios LDPC Codes Tanner Graphs Optimal Cycle-Free Codes LDPC Codes Hard-Decision Decoding Soft-Decision Decoding Irregular LDPC Codes Special LDPC Codes Classification of LDPC Codes Gallager Codes IRA Codes Systematic Codes Turbo Codes BP Decoding Practical Evaluation of LDPC Codes Discrete Distributions Polynomial Interpolation Chernoff Bound Gaussian Distribution Poisson Distribution Degree Distribution Probability Distributions Probability Computation Soliton Distributions Erasure Codes Erasure Codes Tornado Codes Rateless Codes Online Codes Fountain Codes Luby Transform Codes Transmission Methods Luby Transform (LT) Codes Performance Comparison of LT Codes with Other Codes Raptor Codes Evolution of Raptor Codes Importance Sampling Coupon Collector''s Algorithm Open Problems Appendices A ASCII Table B Some Useful Groups C Tables in Finite Fields D Discrete Fourier Transform E Software Resources Bibliography Index ;lt;BR>Extended Hamming Codes SEC-DED Codes Hamming (8,4) Code Hamming (13,8) Code Hamming (32,26) Code Hamming (72,64) Code Hsiao Code Product Notes Uses of Hamming Codes Bounds in Coding Theory Definitions Sphere-Packing Bound Johnson Bound Gilbert-Varshamov Bound Hamming Bound Singleton Bound Plotkin Bound Griesmer Bound Zyablov Bound Bounds in F2n Reiger Bound Krawtchouk Polynomials Linear Programming Bound Stochastic Bounds for SEC-DED Codes Golay Codes Perfect Codes Geometrical Representation Other Construction Methods Finite-State Codes MacWilliams'' Identity Golay''s Original Algorithm Structure of Linear Codes Galois Fields Finite Fields Construction of Galois Fields Galois Fields of Order p Prime Fields Binary Fields Arithmetic in Galois Fields Polynomials Polynomial Codes Matrix Codes Matrix Group Codes Encoding and Decoding Matrices Decoding Procedure Hadamard Code Hadamard Transform Hexacode Lexicodes Octacode Simplex Codes Block Codes Cyclic Codes Definition Construction of Cyclic Codes Methods for Describing Cyclic Codes Quadratic-Residue Codes BCH Codes Binary BCH Codes Extended Finite Fields Construction of BCH Codes General Definition General Algorithm Reed-Muller Codes Boolean Polynomials RM Encoding Generating Matrices for RM Codes Properties of RM Codes Classification of RM Codes Decoding of RM Codes Recursive Definition Probability Analysis Burst Errors Reed-Solomon Codes Definition Reed-Solomon''s Original Approach Parity Check Matrix RS Encoding and Decoding Burst Errors Erasures Concatenated Systems Applications Belief Propagation Rational Belief Belief Propagation Stopping Time Probability Density Function Log-Likelihood Ratios LDPC Codes Tanner Graphs Optimal Cycle-Free Codes LDPC Codes Hard-Decision Decoding Soft-Decision Decoding Irregular LDPC Codes Special LDPC Codes Classification of LDPC Codes Gallager Codes IRA Codes Systematic Codes Turbo Codes BP Decoding Practical Evaluation of LDPC Codes Discrete Distributions Polynomial Interpolation Chernoff Bound Gaussian Distribution Poisson Distribution Degree Distribution Probability Distributions Probability Computation Soliton Distributions Erasure Codes Erasure Codes Tornado Codes Rateless Codes Online Codes Fountain Codes Luby Transform Codes Transmission Methods Luby Transform (LT) Codes Performance Comparison of LT Codes with Other Codes Raptor Codes Evolution of Raptor Codes Importance Sampling Coupon Collector''s Algorithm Open Problems Appendices A ASCII Table B Some Useful Groups C Tables in Finite Fields D Discrete Fourier Transform E Software Resources Bibliography Index ity Golay''s Original Algorithm Structure of Linear Codes Galois Fields Finite Fields Construction of Galois Fields Galois Fields of Order p Prime Fields Binary Fields Arithmetic in Galois Fields Polynomials Polynomial Codes Matrix Codes Matrix Group Codes Encoding and Decoding Matrices Decoding Procedure Hadamard Code Hadamard Transform Hexacode Lexicodes Octacode Simplex Codes Block Codes Cyclic Codes Definition Construction of Cyclic Codes Methods for Describing Cyclic Codes Quadratic-Residue Codes BCH Codes Binary BCH Codes Extended Finite Fields Construction of BCH Codes General Definition General Algorithm Reed-Muller Codes Boolean Polynomials RM Encoding Generating Matrices for RM Codes Properties of RM Codes Classification of RM Codes Decoding of RM Codes Recursive Definition Probability Analysis Burst Errors Reed-Solomon Codes Definition Reed-Solomon''s Original Approach Parity Check Matrix RS Encoding and Decoding Burst Errors Erasures Concatenated Systems Applications Belief Propagation Rational Belief Belief Propagation Stopping Time Probability Density Function Log-Likelihood Ratios LDPC Codes Tanner Graphs Optimal Cycle-Free Codes LDPC Codes Hard-Decision Decoding Soft-Decision Decoding Irregular LDPC Codes Special LDPC Codes Classification of LDPC Codes Gallager Codes IRA Codes Systematic Codes Turbo Codes BP Decoding Practical Evaluation of LDPC Codes Discrete Distributions Polynomial Interpolation Chernoff Bound Gaussian Distribution Poisson Distribution Degree Distribution Probability Distributions Probability Computation Soliton Distributions