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Polynomial theory of error correcting codes
Cancellieri G., Springer Publishing Company, Incorporated, New York, NY, 2014. 732 pp. Type: Book (978-3-319017-26-6)
Date Reviewed: Aug 19 2015

We all know a famous saying: “Mathematics is the queen of science, and number theory is the crown of the queen.” In a similar manner, one can say that coding theory is the queen of information and communication technology (ICT) and cyclic codes are the crown of the queen. They are widely used in many of the modern ICT devices such as CDs, DVDs, mobile phones, networks, and so on. The backbone of the cyclic codes is polynomials. Surprisingly, almost all good codes can be handled through polynomials. This book has identified this beautiful approach of polynomials in studying the theory of error-correcting codes.

This large book--including 12 chapters, four appendices, an index, 250 definitions, 300 propositions (theorems, corollaries, and lemmas), and nearly 500 examples--contains variations of polynomials to combine block codes, convolutional codes, concatenated codes, product codes, and so on under one umbrella. Each chapter has its own references. This book is divided into three parts. Part 1 develops the polynomial approach of the linear error-correcting codes via the generator matrices. It starts with cyclic codes and then considers quasi-cyclic codes, and later focuses on convolutional codes in four chapters. Part 2 studies the same topics through parity check matrices in four chapters. In Part 3, the next four chapters deal with more modern coding theory, such as turbo codes, low-density parity check (LDPC) codes, LDPC convolutional codes, and binomial product generator LDPC block codes.

The book develops new mathematical tools such as interleaved polynomial arithmetic, modified code lengthening, and modified H-extensions. Encoder circuits, state diagrams, and decoding complexity have been discussed for the considered class of codes. A link between direct product codes and convolutional codes is pointed out. A beautiful connection between an H-extension of quasi-cyclic codes and LDPC codes and between cyclic block codes and convolutional codes is given. Many such striking connections arise between different classes of codes due to the polynomial approach. Finally, the four large appendices collect mathematical facts about matrix algebra in a binary finite field, the polynomial representation and arithmetic of binary sequences, electronic circuits in polynomial arithmetic, and a survey on the performance of error correcting codes.

The unified approach of polynomials makes this book quite interesting. It will be quite useful for researchers and engineers working in the field. It can also be used as supplementary material for an advanced undergraduate course or a graduate-level course in coding theory. Mathematicians will certainly like it. The book could have attracted a larger audience if more examples were given at important places after definitions. For example, after Definition 1.8, an example could have been given. Sometimes they are missing. Despite this, the book is good reading. I enjoyed it and I strongly recommend it to any coding theorists.

Reviewer:  Manish Gupta Review #: CR143706 (1511-0935)
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