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Algebraic coding theory : revised edition
Berlekamp E., World Scientific Publishing Co, Inc., River Edge, NJ, 2015. 500 pp. Type: Book
Date Reviewed: Jun 22 2016

Algebraic coding theory has its roots in mathematics and applications in computer science, and its origins in electrical engineering. I was unaware of the area back in 1993, when I joined the PhD program at the Indian Institute of Technology (IIT) Kanpur, so I went to the library and found many books on the subject. The book by Elwyn Berlekamp attracted me immediately as it suits my taste for mathematics with beautiful applications. The first edition came out in 1968, and then a revised edition in 1984. The field has grown tremendously since then and a revised edition was long overdue. A natural question comes to mind: What are the new things in the current revised edition?

I can safely say that this book is still a fundamental reference in the area. It is divided into 16 chapters, three prefaces, two appendices, two references, one index, and one acknowledgement. Chapter 1 starts with a mild introduction to single error-correcting Hamming codes and its natural (the way it was discovered) generalization to double error-correcting BCH codes. This leads to solving equations over finite fields; hence, the next three chapters (3, 4, and 5) build on the necessary algebra of polynomials and finite fields. Incidentally, BCH codes are also cyclic codes, so chapter 5 gives an introduction to binary cyclic codes using the codes of chapter 1. As cyclic codes lead to factorization of polynomial xn-1, chapter 6 discusses this in detail. Chapter 7 provides an introduction to binary BCH codes for correcting multiple errors, a natural generalization of chapter 1. Chapter 8 gives a very naive introduction to non-binary coding. Chapter 9 considers the generalization of cyclic codes (negacyclic codes) with respect to the Lee weight over an odd prime field. Non-binary BCH codes are introduced in chapter 10. This motivates discussions on roots finding algorithms of polynomials over finite fields in chapter 11. The enumeration of information symbols in BCH codes, and its connection to the enumeration to certain kinds of sequences, is discussed in detail in chapter 12. Several beautiful bounds with respect to Hamming and Lee distance are discussed in chapter 13. Several constructions of obtaining good codes using known codes are discussed in chapter 14. Chapter 15 provides an introduction to other useful, well-known classes of block codes and their decoding. Finally, chapter 16 discusses the important concept of weight enumerators (the relationship between weight distributions of code and their duals).

Most of the book’s chapters also provide code implementations (encoding and decoding) and other concepts with respect to circuits, so it is quite useful for engineers. The strong point of the book is its detailed discussions on decoding methods and algorithms for solving corresponding equations over finite fields. The book introduces readers to each concept in a joyful manner.

It is also useful for computer scientists, as most of the algorithms are given in detail with motivations. Obviously, the book is a source of beautiful mathematics, so it always attracts mathematicians. All three types of readers must have this book; I strongly recommend it.

Reviewer:  Manish Gupta Review #: CR144523 (1609-0630)
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