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Geometry and codes. Transl. from the Russian by N. G. Shartse. (English) Zbl 1097.14502
Mathematics and Its Applications. Soviet Series 24. Dordrecht etc.: Kluwer Academic Publishers (ISBN 90-277-2776-7). ix, 157 p. (1988).
The history of error-correcting codes began in 1948 with the publication of a famous paper by C. Shannon. Hamming introduced a class of single-error-correcting block codes in 1950. Ten years later BCH codes and Reed-Solomon codes were discovered. In 1970 the author defined a new class of linear codes in terms of rational functions instead of hitherto used polynomials. These codes are nowadays called classical Goppa codes. In 1977 the author proposed the idea of applying the Riemann-Roch theorem in algebraic geometry to the analysis of codes. In subsequent papers [Math. USSR, Izv. 21, 75–91 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, 762–781 (1982; Zbl 0522.94013); Russ. Math. Surv. 39, No. 1, 87–141 (1984); translation from Usp. Mat. Nauk 39, No. 1(235), 77–120 (1984; Zbl 0578.94011)], he developed the above idea and clarified the relation between the theory of error-correcting codes and the theory of algebraic curves defined over finite fields.
The present book expounds these results and ideas. It consists of four chapters: 1. Rational codes; 2. Decoding and rational approximations; 3. Algebraic curves; 4. Generalized Jacobian codes.
In Chapter 1, the classical linear codes are presented as codes on a projective line, ovals or rational algebraic curves. Chapter 2 treats some aspects of decoding, rational approximations, shift registers and linear discrete filters. Chapter 3 provides a compact introduction to the theory of algebraic curves defined over the finite field $$\mathbb F_q$$. Bézout’s theorem, the Riemann-Roch theorem and the Hasse-Weil bound are explained, for the most part without proofs. Chapter 4 introduces Jacobian and generalized Jacobian varieties of an algebraic curve and their associated codes.
Finally, Fermat and Hermitian curves as well as modular curves are defined and some results are discussed. The list of references contains 16 titles, of which only two are classical papers on error-correcting codes and others are treatises on algebraic geometry or number theory. The author writes in the preface that the book does not presuppose the reader’s familiarity with coding theory or algebraic geometry. But in order to appreciate this book, it is desirable that the reader have a fairly thorough knowledge of both these domains.
The author is the founder of the theory of algebro-geometric codes and this monograph is the first systematic treatise of this new theory written by the author himself.
The original text was written in Russian. The English translation is good enough for one to understand the mathematical contents. But several mistranslations are found here and there. For example “specialization index” (p. 124) should be “specialty index”, ”arbitrary point” might mean ”generic point” in the sense of A. Weil [ Foundations of algebraic geometry, Am. Math. Soc., New York, 1962; Zbl 0168.18701)], “principle divisors” (pp. 116, 126) may be “principal divisors” and so on.

##### MSC:
 14G50 Applications to coding theory and cryptography of arithmetic geometry 14G15 Finite ground fields in algebraic geometry 11T71 Algebraic coding theory; cryptography (number-theoretic aspects) 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 94-02 Research exposition (monographs, survey articles) pertaining to information and communication theory 94Bxx Theory of error-correcting codes and error-detecting codes