Polynomial theory of error correcting codes (Q483340)

The goal of this book is to develop a polynomial theory for several classes of codes including block and convolutional codes. The book has 732 pages and the treatment of the theory is well done from the engineering perspective. The well-structured book contains three main parts and four appendices. The first part focuses on generator matrices for linear block codes, wide-sense time-invariant block codes, strict-sense time-invariant convolutional codes, and wide-sense time-invariant convolutional codes. Chapter 1 gives a polynomial description of a linear block code. In particular, classical cyclic codes, and several different variations including direct product codes have been addressed. The systematic encoder circuit and \(Q\)-ary state diagrams are discussed as well. Wide-sense time-invariant block codes and their polynomial interleaved products are the topic of Chapter 2. As a generalization of cyclic codes, quasi-cyclic codes and their variations are treated in details. Chapters 3 and 4 deal with strict-sense and wide-sense time-invariant convolutional codes respectively. Bridges between block codes and convolutional codes are demonstrated. Part II contains chapters 5-8 which concentrate on parity check matrices. An analogous but distinct theory for these block and convolutional codes are developed through these four chapters. For example, the polynomial approach to the modified H-extended block codes are described in chapters 5 and 6; another bridge between block codes and convolutional codes based on the parity check matrix is also established in chapters 7 and 8. Finally, part III contains four chapters aiming to offer an comprehensive understanding of the structures of two modern types of codes: Turbo codes and LDPC codes. While Chapters 9 and 10 provide the basic introduction to these codes, Chapters 11--12 focus on binomial product generator LDPC block codes and LDPC convolutional codes respectively. Four appendices cover topics such as matrix algebra in a binary finite field, polynomial representation of binary sequences, electronic circuits for multiplication/division of polynomial representation of binary sequences, and the main performance of error correcting codes. These appendices provide useful background to general readers. In summary, this well-written book offers a structural and comprehensive view of the polynomial theory of many classes of error correcting codes to a general reader. In particular, nearly \(500\) examples and many nice figures are very helpful.