The theory of codes is at the crossroads of abstract algebra, applied mathematics and computer science. The increasing need for codes, propelled by the incredible growth of the world wide web, attracts the interest of students, theorists and professionals of the world of information. Now, more than ever, we need thorough introductory textbooks to satisfy both the needs of undergraduate students who are about to learn the basics of linear codes, and the needs of more advanced students taking an advanced course on abstract coding theory.
Introduction to Coding Theory is a ample source of information on the subject (566 pages), from the basic definitions to advanced topics. The volume is intended for upper level undergraduates in Computer Science, Electrical Engineering and Mathematics. Indeed, some knowledge of linear algebra, abstract algebra, probability and combinatorics is assumed (and needed).
The first four chapters go through the basics of communication systems, block codes, linear codes, bounds on the parameters of codes and a fairly extensive introduction to the properties of finite fields (which are treated in more detail in a later chapter). The following four chapters introduce the encoding and decoding of Reed-Solomon codes and other similar codes. Cyclic codes and Bose-Chaudhuri-Hocquenghem (BCH) codes are also treated in some detail. The later and more advanced chapters deal with list decoding of Reed-Solomon codes, codes in the Lee metric, maximum distance separable MDS codes, concatenated codes, graph codes and trellis and convolutional codes. The appendix of the book is a brief review of the basics of modern algebra.
In my opinion, the book is a nicely written, comprehensive introduction to coding theory. I really appreciate the fact that the volume seems intended not just as a textbook for a first course in coding theory, but rather as a book that can be used in several courses at different levels, and as a useful resource for the reader. The book provides a large number of references (over 400) to other books and research articles in the subject, for those who want to study deeper aspects not covered in the volume. Moreover, even though the book is intended for undergraduate students in several fields, the mathematical rigor has been kept intact. There are also many problems at the end of each section, as well as notes about the computational aspects of the theory. My only concern about the book is the appendix on modern algebra which, in the present edition, seems too short (6 pages) to serve any useful purpose. Perhaps the appendix should be expanded in the next edition of the book.
Álvaro Lozano-Robledo is H. C. Wang Assistant Professor at Cornell University.