Finite fields for computer scientists and engineers.

*(English)*Zbl 0662.94014
The Kluwer International Series in Engineering and Computer Science. Information Theory, 23. Boston-Dordrecht-Lancaster: Kluwer Academic Publishers. X, 207 p.; Dfl. 105.00; $ 39.95; £29.95 (1987).

This slim volume contains an interesting collection of topics in the area of finite fields and their applications, some of which have not previously appeared in text book form.

The first six chapters on properties of finite fields are as might be found in many books on coding theory, with perhaps more emphasis here on Euclidean domains than is common. The essential properties of such fields are established in a logical and consistent manner.

Chapter 7 considers the Berlekamp algorithm for the fatorization of polynomials. It begins with a fairly detailed discussion of cyclotomic polynomials as a useful first step in the factorization of a polynomial.

The following chapter considers the bit serial multiplication algorithm, also due to Berlekamp, and to this end discusses the properties of the trace and norm functions, solutions of quadratic equations and the notion of dual bases and their use in such multiplication algorithms.

The last three chapters consider the subject of linear recurrences, linear feedback shift register sequences (called m-sequences here) and cross correlation properties of certain classes of sequences, topics made more important by the advent of spread spectrum and multi-user communication systems. The last chapter on cross correlations is perhaps somewhat deeper than other sections of the book and gives an interesting and readable account of the topic. Each chapter comes with a useful collection of problems.

The first six chapters on properties of finite fields are as might be found in many books on coding theory, with perhaps more emphasis here on Euclidean domains than is common. The essential properties of such fields are established in a logical and consistent manner.

Chapter 7 considers the Berlekamp algorithm for the fatorization of polynomials. It begins with a fairly detailed discussion of cyclotomic polynomials as a useful first step in the factorization of a polynomial.

The following chapter considers the bit serial multiplication algorithm, also due to Berlekamp, and to this end discusses the properties of the trace and norm functions, solutions of quadratic equations and the notion of dual bases and their use in such multiplication algorithms.

The last three chapters consider the subject of linear recurrences, linear feedback shift register sequences (called m-sequences here) and cross correlation properties of certain classes of sequences, topics made more important by the advent of spread spectrum and multi-user communication systems. The last chapter on cross correlations is perhaps somewhat deeper than other sections of the book and gives an interesting and readable account of the topic. Each chapter comes with a useful collection of problems.

Reviewer: I.F.Blake

##### MSC:

94B05 | Linear codes, general |

11T06 | Polynomials over finite fields |

11T99 | Finite fields and commutative rings (number-theoretic aspects) |

94-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory |

12-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory |