Elements of Algebraic Coding Systems

Description

Elements of Algebraic Coding Systems is an introductory text to algebraic coding theory. In the first chapter, you'll gain inside knowledge of coding fundamentals, which is essential for a deeper understanding of state-of-the-art coding systems. This book is a quick reference for those who are unfamiliar with this topic, as well as for use with specific applications such as cryptography and communication. Linear error-correcting block codes through elementary principles span eleven chapters of the text. Cyclic codes, some finite field algebra, Goppa codes, algebraic decoding algorithms, and applications in public-key cryptography and secret-key cryptography are discussed, including problems and solutions at the end of each chapter. Three appendices cover the Gilbert bound and some related derivations, a derivation of the Mac- Williams' identities based on the probability of undetected error, and two important tools for algebraic decoding--namely, the finite field Fourier transform and the Euclidean algorithm for polynomials Content: 1. Basic concepts -- 1.1 Introduction -- 1.2 Types of errors -- 1.3 Channel models -- 1.4 Linear codes and non-linear codes -- 1.5 Block codes and convolutional codes -- 1.6 Problems with solutions. 2. Block codes -- 2.1 Introduction -- 2.2 Matrix representation -- 2.3 Minimum distance -- 2.4 Error syndrome and decoding -- 2.4.1 Maximum likelihood decoding -- 2.4.2 Decoding by systematic search -- 2.4.3 Probabilistic decoding -- 2.5 Simple codes -- 2.5.1 Repetition codes -- 2.5.2 Single parity-check codes -- 2.5.3 Hamming codes -- 2.6 Low-density parity-check codes -- 2.7 Problems with solutions. 3. Cyclic codes -- 3.1 Matrix representation of a cyclic code -- 3.2 Encoder with n -- k shift-register stages -- 3.3 Cyclic Hamming codes -- 3.4 Maximum-length-sequence codes -- 3.5 Bose-Chaudhuri-Hocquenghem codes -- 3.6 Reed-Solomon codes -- 3.7 Golay codes -- 3.7.1 The binary (23, 12, 7) Golay code -- 3.7.2 The ternary (11, 6, 5) Golay code -- 3.8 Reed-Muller codes -- 3.9 Quadratic residue codes -- 3.10 Alternant codes -- 3.11 Problems with solutions. 4. Decoding cyclic codes -- 4.1 Meggitt decoder -- 4.2 Error-trapping decoder -- 4.3 Information set decoding -- 4.4 Threshold decoding -- 4.5 Algebraic decoding -- 4.5.1 Berlekamp-Massey time domain decoding -- 4.5.2 Euclidean frequency domain decoding -- 4.6 Soft-decision decoding -- 4.6.1 Decoding LDPC codes -- 4.7 Problems with solutions. 5. Irreducible polynomials over finite fields -- 5.1 Introduction -- 5.2 Order of a polynomial -- 5.3 Factoring xqn -- x -- 5.4 Counting monic irreducible q-ary polynomials -- 5.5 The Moebius inversion technique -- 5.5.1 The additive Moebius inversion formula -- 5.5.2 The multiplicative Moebius inversion formula -- 5.5.3 The number of irreducible polynomials of degree n over GF(q) -- 5.6 Chapter citations -- 5.7 Problems with solutions. 6. Finite field factorization of polynomials -- 6.1 Introduction -- 6.2 Cyclotomic polynomials -- 6.3 Canonical factorization -- 6.4 Eliminating repeated factors -- 6.5 Irreducibility of ̲[phi]n(x) over GF(q) -- 6.6 Problems with solutions. 7. Constructing f-reducing polynomials -- 7.1 Introduction -- 7.2 Factoring polynomials over large finite fields -- 7.2.1 Resultant -- 7.2.2 Algorithm for factorization based on the resultant -- 7.2.3 The Zassenhaus algorithm -- 7.3 Finding roots of polynomials over finite fields -- 7.3.1 Finding roots when p is large -- 7.3.2 Finding roots when q = pm is large but p is small -- 7.4 Problems with solutions. 8. Linearized polynomials -- 8.1 Introduction -- 8.2 Properties of L(x) -- 8.3 Properties of the roots of L(x) -- 8.4 Finding roots of L(x) -- 8.5 Affine q-polynomials -- 8.6 Problems with solutions. 9. Goppa codes -- 9.1 Introduction -- 9.2 Parity-check equations -- 9.3 Parity-check matrix of Goppa codes -- 9.4 Algebraic decoding of Goppa codes -- 9.4.1 The Patterson algorithm -- 9.4.2 The Blahut algorithm -- 9.5 The asymptotic Gilbert bound -- 9.6 Quadratic equations over GF(2m) -- 9.7 Adding an overall parity-check digit -- 9.8 Affine transformations -- 9.9 Cyclic binary double-error correcting -- 10. Extended Goppa codes -- 9.10 Extending the Patterson algorithm for decoding Goppa codes -- 9.11 Problems with solutions. 10. Coding-based cryptosystems -- 10.1 Introduction -- 10.2 McEliece's public-key cryptosystem -- 10.2.1 Description of the cryptosystem -- 10.2.2 Encryption -- 10.2.3 Decryption -- 10.2.4 Cryptanalysis -- 10.2.5 Trapdoors -- 10.3 Secret-key algebraic coding systems -- 10.3.1 A (possible) known-plaintext attack -- 10.3.2 A chosen-plaintext attack -- 10.3.3 A modified scheme -- 10.4 Problems with solutions. 11. Majority logic decoding -- 11.1 Introduction -- 11.2 One-step majority logic decoding -- 11.3 Multiple-step majority logic decoding I -- 11.4 Multiple-step majority logic decoding II -- 11.5 Reed-Muller codes -- 11.6 Affine permutations and code construction -- 11.7 A class of one-step decodable codes -- 11.8 Generalized Reed-Muller codes -- 11.9 Euclidean geometry codes -- 11.10 Projective geometry codes -- 11.11 Problems with solutions. Appendices -- A. The Gilbert bound -- A.1. Introduction -- A.2. The binary asymptotic Gilbert bound -- A.3. Gilbert bound for linear codes -- B. MacWilliams' identity for linear codes -- B.1. Introduction -- B.2. The binary symmetric channel -- B.3. Binary linear codes and error detection -- B.4. The q-ary symmetric channel -- B.5. Linear codes over GF(q) -- B.6. The binomial expansion -- B.7. Digital transmission using N regenerative repeaters -- C. Frequency domain decoding tools -- C.1. Finite field Fourier transform -- C.2. The Euclidean algorithm.