Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- 1 Introduction -- Exercises -- 2 Error detection, correction and decoding -- 2.1 Communication channels -- Decoding rule -- 2.2 Maximum likelihood decoding -- 2.3 Hamming distance -- 2.4 Nearest neighbour/minimum distance decoding -- 2.5 Distance of a code -- Exercises -- 3 Finite fields -- 3.1 Fields -- 3.2 Polynomial rings -- 3.3 Structure of finite fields -- 3.4 Minimal polynomials -- Exercises -- 4 Linear codes -- 4.1 Vector spaces over finite fields -- 4.2 Linear codes -- 4.3 Hamming weight -- 4.4 Bases for linear codes -- 4.5 Generator matrix and parity-check matrix -- 4.6 Equivalence of linear codes -- 4.7 Encoding with a linear code -- 4.8 Decoding of linear codes -- 4.8.1 Cosets -- 4.8.2 Nearest neighbour decoding for linear codes -- 4.8.3 Syndrome decoding -- Exercises -- 5 Bounds in coding theory -- 5.1 The main coding theory problem -- 5.2 Lower bounds -- 5.2.1 Sphere-covering bound -- 5.2.2 Gilbert-Varshamov bound -- 5.3 Hamming bound and perfect codes -- 5.3.1 Binary Hamming codes -- Decoding with a binary Hamming code -- 5.3.2 q-ary Hamming codes -- Decoding with a q-ary Hamming code -- 5.3.3 Golay codes -- Binary Golay codes -- Ternary Golay codes -- 5.3.4 Some remarks on perfect codes -- 5.4 Singleton bound and MDS codes -- 5.5 Plotkin bound -- 5.6 Nonlinear codes -- 5.6.1 Hadamard matrix codes -- 5.6.2 Nordstrom-Robinson code -- 5.6.3 Preparata codes -- 5.6.4 Kerdock codes -- 5.7 Griesmer bound -- 5.8 Linear programming bound -- Exercises -- 6 Constructions of linear codes -- 6.1 Propagation rules -- 6.2 Reed-Muller codes -- 6.3 Subfield codes -- Exercises -- 7 Cyclic codes -- 7.1 Definitions -- 7.2 Generator polynomials -- 7.3 Generator and parity-check matrices -- 7.4 Decoding of cyclic codes -- Decoding algorithm for cyclic codes.

Decoding algorithm for cyclic burst-error-correcting codes -- 7.5 Burst-error-correcting codes -- Exercises -- 8 Some special cyclic codes -- 8.1 BCH codes -- 8.1.1 Definitions -- 8.1.2 Parameters of BCH codes -- 8.1.3 Decoding of BCH codes -- 8.2 Reed-Solomon codes -- 8.3 Quadratic-residue codes -- Exercises -- 9 Goppa codes -- 9.1 Generalized Reed-Solomon codes -- 9.2 Alternant codes -- 9.3 Goppa codes -- 9.4 Sudan decoding for generalized RS codes -- 9.4.1 Generation of the (P, k, t)-polynomial -- 9.4.2 Factorization of the (P, k, t)-polynomial -- Factoring algorithm -- Exercises -- References -- Bibliography -- Index.