Error-correcting codes and finite fields /

Error-correcting codes are a means of securing data, for the purposes of transmission or storage, by adding redundancy. Satellite photographs, digital audio, and fiber-optic telephone lines all depend on such codes. Many of these codes in turn depend on the algebra and geometry of finite fields, once the exclusive purview of the pure mathematician. Indeed, coding theory is an attractive topic for the undergraduate curriculum precisely because it enlivens abstract algebra for the practically minded. Now, a code's efficiency reflects the trade-off between the reliability of the data and the added length of the message, and constructing more efficient codes is enormously important. In 1970, V.D. Goppa invented a class of strikingly efficient codes using algebraic geometry. The feature that distinguishes the present volume among the many good works on coding theory is that it culminates with an undergraduate-level introduction to Goppa codes using a minimum of theoretical baggage. (V.P. Goppa's own Geometry and Codes, 1988, is good reading but more advanced; M.A. Tsfasman and S.G. Vladut's encyclopedic Algebraic-Geometric Codes, 1991, is for now the definitive reference). Indeed, all facts about finite fields are developed ab initio, though important facts from algebraic geometry must be quoted without proof. The exposition is clear and leisurely throughout, with many worked examples. Undergraduate. D. V. Feldman; University of New Hampshire