Shi, Minjia; Alahmadi, Adel; SolĂ©, Patrick Codes and rings. Theory and practice. (English) Zbl 1386.94002 Pure and Applied Mathematics. Amsterdam: Elsevier/Academic Press (ISBN 978-0-12-813388-0/pbk; 978-0-12-813391-0/ebook). xv, 301 p. (2017). Rings play two important roles in coding theory firstly as alphabets, secondly as an underlying structure for algebraic codes. This book is a research survey of literature that focuses on codes over rings and rings acting on codes. It is summarizing the last 20–25 years of research and applications on the subject “Codes and Rings: Theory and Practice” on 301 pages.The book is divided into 14 chapters.1. Motivation, 2. Rings, 3. Distances, 4. Few weight codes, 5. Linear codes, 6. Self-dual codes, 7. Cyclic codes, 8. Quasicyclic codes, 9. Quasitwisted codes, 10. Generalized quasicyclic codes, 11. Skew cyclic codes, 12. MDR codes, 13. Convolutional codes, 14. Character sums.The authors of this book are experts of the coding theory, ring theory and related fields. Since the topic is very large, the authors inevitably had to omit several themes, for example the theory of codes over rings which requires deep background in algebraic geometry. A detailed collection of basic references, including the authors’ results, are added in each chapter.The book contains examples and exercises to illustrate the results and to understand the main themes of the book.Rings associated with codes in this book are assumed to be finite, but in some cases infinite rings are also allowed. Commutativity of the rings is not always assumed as in some applications, matrix rings, skew polynomial rings, are included. The book contains new results by the authors that complement existing theories. Such areas are skew cyclic codes, decomposition theory of quasicyclic and other generalized cyclic codes under ring action, MDS convolutional codes over finite rings, duality over Frobenius rings.The authors point out how the subject is related to other fields e.g. combinatorics, number theory and ring theory. Engineering applications of codes over rings are also outlined. Such applications are low correlation sequences, Euclidean lattices, and combinatorial designs, space-time coding, various kinds of pseudo-random sequences. Since the subject is still in the stage of development the authors give some motivation points, open problems.The book is very useful for references and can be also be used for the serious study of the subject. Ring theorists principally at PhD and research level, coding theorist interested in applications (computer scientists and cryptologists) could appreciate the collection and arrangement of research content of more than 250 references. Reviewer: Piroska Lakatos (Debrecen) Cited in 1 ReviewCited in 23 Documents MSC: 94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory 94B05 Linear codes (general theory) 94B15 Cyclic codes 94B40 Arithmetic codes 94A60 Cryptography 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 13M05 Structure of finite commutative rings Keywords:linear codes; rings; Frobenius rings; chain rings; Galois rings; distances; Lee; homogeneous and Hamming metric; few weight codes; self-dual codes; cyclic codes; quasi-cyclic codes; quasi-twisted codes; generalized quasi-cyclic codes; skew cyclic codes; MDR codes; MDS codes; convolutional codes; character sums; Gilbert-Varshamov bound; Singleton bound PDFBibTeX XMLCite \textit{M. Shi} et al., Codes and rings. Theory and practice. Amsterdam: Elsevier/Academic Press (2017; Zbl 1386.94002) Full Text: Link