Modern communication and information systems are based on the reliable yet efficient transmission of information. Communication channels are plagued by noise and are thus susceptible to errors, regardless of whether they involve transmission over noisy radio channels or transmission on optical storage devices damaged by scratches. Nowadays, communication and information systems can be designed to reduce transmission errors to arbitrarily small probabilities thanks to the theoretical work over the past 65 years that produced coding theory. Classical coding theory uses algebraic methods, while modern coding theory turns more to statistics and probability theory. Not many textbooks cover these more recent developments in coding theory.

In this book, topics are organized chronologically, starting with a short history of codes that preceded classical coding theory. Chapter 2 reviews certain aspects of digital arithmetic that apply to the generation of error detecting and error correcting codes. The beginning of classical coding theory is marked by the invention of linear codes, in particular, Hamming codes, and the associated theory. Chapter 3 reviews the mathematical and information theoretical preliminaries that lead to classical coding theory. A whole chapter is then devoted to Hamming codes and another one to extended Hamming codes, which are able to correct single errors or detect double errors. Often in applied mathematics, as in complexity theory, information theory, and coding theory, only the theoretical bounds can be determined, whereas the algorithms to reach those bounds remain unknown. Chapter 6 presents all known bounds of coding theory. Such a comprehensive collection of bounds is rather unique. In chapter 7, perfect linear codes are discussed, with their most important representatives, the Golay codes, which were used, for example, on the Voyager 1 and 2 spacecrafts for deep space communication. Chapter 8 presents the next phase in the development of classical coding theory: Galois field theory and the associated arithmetic. In chapter 9, the authors discuss matrix codes, in particular, Hadamard codes, which play an important role in modern communication systems. Applications of Galois field theory are next: “cyclic codes in chapter 10, BCH codes in chapter 11, Reed-Muller codes in chapter 12, and Reed-Solomon codes in chapter 13.”

The exploration of modern coding theory begins with the introduction of belief propagation and stochastic processes. The belief propagation algorithm is an efficient method to solve inference problems by logical means. Chapter 14 does not reveal how belief propagation can be used in coding theory. However, in chapter 15, along with the presentation of low-density parity-check (LDPC) codes, the practical use of belief propagation becomes apparent. Special LDPC codes, in particular Gallager codes, are examined in chapter 16. Discrete distributions, one of the cornerstones of modern coding theory, are presented in chapter 17. Chapters 18 through 20 discuss four recently developed codes that cannot easily be found in other textbooks, namely the Tornado, Fountain, Luby Transform, and Raptor codes. The first two are discussed in chapter 18, and the latter two in chapters 19 and 20. Finally, five appendices cover the well-known ASCII table, some useful tables on finite and Galois fields, a short essay on the discrete Fourier transform, and a listing of freely available online resources. The book contains about 200 examples, but no exercises. The illustrations and tables are well chosen and help the reader comprehend the often-abstract subject matter.

Coding theory is a large and still developing field that cannot be treated completely in a single book. For example, space-time codes, which are of practical importance in modern communication systems, have been omitted. But overall, the authors managed to cover the most important parts of and newest developments in coding theory in a terse but still comprehensible manner. I strongly recommend the book to graduate students and professionals in communications engineering.