Theory and Practice of Error Control Codes

alternant codes BCH code Berlekamp-Massey algorithm binary code block codes burst error C₁ called channel circuit code of blocklength code over GF(q codeword coefficients columns components compute conjugacy constraint length construct convolutional code correct cyclic code cyclic convolution decoding algorithm defined degree denoted designed distance elements of GF(q encoder erasures error pattern error-control codes Euclidean example field element field GF(q Figure Fourier transform Galois field GF(p Goppa code GRM code Hamming code Hence information bits information symbols input integer inverse iterations linear code majority decoder minimal polynomial minimum distance modulo multiplication node nomial nonzero parity frequencies parity-check equations parity-check matrix permutation polynomial g(x primitive element Proof prove q-ary quadratic residue received word recursive Reed-Muller code Reed-Solomon code relatively prime ring S₁ satisfies sequence shift register shown in Fig spectrum subspace syndrome two-dimensional V₁ vector space weight