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Book review

Author(s) Tsfasman, M.A.
Vladut, S.G.
Title Algebraic-geometric codes
Publisher Kluwer Academic Publishers
Year of publication 1991
Reviewed by Mirela Stefanescu

The beginning of coding theory was made by C. Shannon's paper on the mathematical theory of communication in 1948. But most of the theory is concerned with codes. A code C is a subset of Q^n where Q is a finite set called alphabet. The vectors in C are called codewords, n being their length. Sending a codeword c over a noisy channel to a receiver, the received word r may differ from c in a number of places, and differences are errors in the received message. If <formula> where <formula>, and <formula>, then the code C is called an e-error correcting code. A linear code is a code with Q = Fq, a finite field. Let F be the algebraic closure of Fq and let X be the projective line over F. The rational points on X are <formula> and <formula>.

Denote by L the space of rational functions on X, over Fq, having sense in all Pi and a pole of order less than k in Q (eventually no pole) and no other poles on X. To each <formula>, it corresponds a codeword <formula> and one obtains the Rees-Solomon code. In this way, a vector space of rational functions defined on the projective line is mapped to a linear code by taking values of the functions at a specified set of points of the line to be the coordinates of the words.

The authors have their own contributions in developing of algebraic-geometric code theory. Their book begins with an introduction to coding theory; that means some elementary theory, examples of codes, bounds. The second big chapter of the book contains an introduction to the theory of curves in algebraic geometry. It is not an elementary introduction, since the authors will apply really the algebraic geometry to coding theory for calculating bounds on minimum distances of these linear codes. They give an account on AG-constructions and properties, decoding and asymptotic results.

The fourth chapter is devoted to studying modular codes (on classical modular curves, on Drinfeld curves) and their complexity (which is polynomial).

The last chapter uses sphere packings for introducing analogues of AG-codes. First they present the concept of sphere packing and asymptotic dense packing.

Each chapter ends with historical and bibliographic notes and the book itself ends with an appendix: summary of results and tables .

The book is a valuable text for specialists both in algebraic geometry and in coding theory and, in spite of the high price, it could be useful to them, adding some new things to the beautiful book published by J.H. van Lint and G. van der Geer in Birkhäuser Verlag, in 1988. Some care for using a correct English language would make the book nicer.