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Codes and rings: theory and practice
Shi M., Alahmadi A., Solé P., Academic Press, London, UK, 2017. 318 pp.  Type: Book (978-0-128133-88-0)
Date Reviewed: Mar 29 2018

Error-correcting codes are everywhere and play an important role in our daily lives, from scanning barcodes or participating in a Skype chat, a credit card transaction, or a phone call, to storing data on computers. They provide data reliability in all forms of information storage and communications. Usually the alphabet of codes used is finite fields; however, certain codes over finite fields have connections to finite rings, for example, as an ideal in polynomial rings and as an easy construction for the best nonlinear codes over finite fields from codes over finite rings. The latter connection was reported to Science magazine [1] (it also resulted in an award-winning paper [2] and Patrick Solé was part of the discovery) when discovered back in the early 1990s as it solved the mysterious connection between two famous binary nonlinear codes, the Kerdock and Preparata codes. Now, more than two decades later, several applications of codes over finite rings have been discovered and the theory has been developed in many directions. This book collects some basic results of codes over finite rings in 14 chapters.

Chapter 1 starts with a motivation, primarily focusing on sequences and lattices. Chapter 2 gives the basics of rings considered in the book such as local rings, Galois rings, chain rings, skew polynomial rings, and Frobenius rings. The famous isometry (Gary map) from codes over Z4 to Z22 preserves the Lee weight to Hamming weight and in general the isometry preserves the homogeneous weight to Hamming weight. Thus, chapter 3 discusses the homogeneous, Lee, and Hamming distances, and provides the basic bounds of coding theory using the generating functions. The small weight codes over finite rings give rise to the strongly regular graphs. One-weight and two-weight codes are considered in chapter 4. Linear code and its dual are studied in chapter 5. Special focus is given to modular independence. The beautiful theory of self-dual codes is studied in chapter 6 with a focus on type II codes. Cyclic codes are covered in chapter 7 by studying the divisors of xn-1. San Ling and Patrick Solé gave fundamental results about quasi-cyclic (generalization of cyclic) codes in a recent series of papers [3,4,5,6]. Chapter 8 discusses results of these papers for finite fields and chain rings. Further generalization of cyclic codes to quasi-twisted, generalized quasi-cyclic (with a focus on LCD codes) and skew cyclic codes (with applications to lattices) are studied in chapters 9, 10, and 11. Codes meeting the Singleton type bound for rank are studied in chapter 12, with a few results on the p-dimension of the code. The famous convolutional codes analogues for finite rings are considered in chapter 13 with background on the p-basis for codes over Zps. Finally, chapter 14 considers character sums (Gauss sums over local rings and Weil sums).

I enjoyed this delightful book and recommend it to young students who want to work on codes over finite rings and their applications. This book will provide a very good foundation for those who want to work in quantum-safe cryptosystems or quantum coding theory, or on any algebraic coding theory topic. It will also give insight to many aspects of the area. The popularity of the book is evident from the fact that most of my own PhD students want to borrow it from me. And one of the co-authors is Patrick Solé, a master of the subject, who has contributed greatly in the area.

Reviewer:  Manish Gupta Review #: CR145940 (1806-0276)
1) Cipra, B. Nonlinear codes straighten up—and get to work. Science 262, 5134(1993), 658–659.
2) Hammons, A. R.; Kumar, P. V.; Calderbank, A. R.; Sloane, N. J. A. The Z4-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Transactions on Information Theory 40, 2(1994), 301–319.
3) Ling, S.; Solé, P. On the algebraic structure of quasi-cyclic codes I: finite fields. IEEE Transactions on Information Theory 47, 7(2001), 2751–2760.
4) Ling, S.; Solé, P. On the algebraic structure of quasi-cyclic codes II: chain rings. Designs, Codes and Cryptography 30, 1(2003), 113–130.
5) Ling, S.; Solé, P. On the algebraic structure of quasi-cyclic codes III: generator theory. IEEE Transactions on Information Theory 51, 7(2005), 2692–2700.
6) Ling, S.; Niederreiter, H.; Solé, P. On the algebraic structure of quasi-cyclic codes IV: repeated roots. Designs, Codes and Cryptography 38, 3(2006), 337–361.
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