The mathematical theory of coding.

*(English)*Zbl 0318.94009
New York - San Francisco - London: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers. xi, 356 p. $ 28.00 (1975).

The authors write in their preface: “The subject of coding theory, for both discrete and continuous channels, has developed rapidly over the past twenty five years with the application of more and more diverse algebraic and combinatorial methods. The aim of this book is to present a unified treatment of these mathematical ideas and their use in the construction of codes. It is not at all concerned with the practical matters of code implementation, and the subject of decoding is considered only insofar as it relates to the mathematical ideas involved. In many instances we have purposely chosen for a problem an approach that is mathematically more advanced than required in order to expose the reader to as wide a scope of concepts as possible, with., in the context of coding. Such an approach is not designed to achieve a given result with maximum efficiency.”

The first three chapters deal with coding for the discrete channel. The first chapter (Finite fields and coding theory) contains an introduction to finite fields together with properties of polynomials and vector spaces over finite fields. Linear codes, cyclic codes, BCH codes, GRM codes, the group of a code are discussed. The second chapter (Combinatorial constructions and coding) presents various combinatorial structures and related codes. Here, for example, the finite geometry codes, further balanced incomplete block designs, Latin squares and Steiner triple systems, quadratic residues, Hadamard matrices, difference sets and related codes are studied. In the third chapter (Coding and combinatorics) connections of combinatorial structures with codes (\(t\)-designs and perfect codes, nearly perfect codes, balanced codes, equidistant codes) are discussed. In the fourth chapter the structure of semisimple rings is explored. This material is common to coding for both discrete and continuous channels. This chapter also serves as background for the study of group representations in the chapter 5. Chapter 6 (Group codes for the Gaussian channel) utilizes this theory for the construction of codes for the Gaussian channel.

At the end of each chapter is a section for comments and references. In addition there are many interesting exercises. The more difficult problems contain references. At the end of the book is an extensive set of references.

This excellent book covers the important developments in algebraic coding theory until the most recent times.

The first three chapters deal with coding for the discrete channel. The first chapter (Finite fields and coding theory) contains an introduction to finite fields together with properties of polynomials and vector spaces over finite fields. Linear codes, cyclic codes, BCH codes, GRM codes, the group of a code are discussed. The second chapter (Combinatorial constructions and coding) presents various combinatorial structures and related codes. Here, for example, the finite geometry codes, further balanced incomplete block designs, Latin squares and Steiner triple systems, quadratic residues, Hadamard matrices, difference sets and related codes are studied. In the third chapter (Coding and combinatorics) connections of combinatorial structures with codes (\(t\)-designs and perfect codes, nearly perfect codes, balanced codes, equidistant codes) are discussed. In the fourth chapter the structure of semisimple rings is explored. This material is common to coding for both discrete and continuous channels. This chapter also serves as background for the study of group representations in the chapter 5. Chapter 6 (Group codes for the Gaussian channel) utilizes this theory for the construction of codes for the Gaussian channel.

At the end of each chapter is a section for comments and references. In addition there are many interesting exercises. The more difficult problems contain references. At the end of the book is an extensive set of references.

This excellent book covers the important developments in algebraic coding theory until the most recent times.

Reviewer: Herbert Pieper (Berlin)