Curves with many points. I: Modular curves
Class field theory
Curves with many points. II
Infinite global fields
Decoding: Some examples
Codes from multidimensional varieties
Appendix: Some basic facts from Volume 1
List of names
Algebraic Geometry Codes: Advanced Chapters is devoted to the theory of algebraic geometry codes, a subject related to several domains of mathematics. On one hand, it involves such classical areas as algebraic geometry and number theory; on the other, it is connected to information transmission theory, combinatorics, finite geometries, dense packings, and so on. The book gives a unique perspective on the subject. Whereas most books on coding theory start with elementary concepts and then develop them in the framework of coding theory itself within, this book systematically presents meaningful and important connections of coding theory with algebraic geometry and number theory. Among many topics treated in the book, the following should be mentioned: curves with many points over finite fields, class field theory, asymptotic theory of global fields, decoding, sphere packing, codes from multi-dimensional varieties, and applications of algebraic geometry codes.
The book is the natural continuation of Algebraic Geometric Codes: Basic Notions by the same authors. The concise exposition of the first volume is included as an appendix.
• Michael Tsfasman, CNRS, Laboratoire de Mathematiques de Versailles, France, Institute for Information Transmission Problems, Moscow, Russia, and Independent University of Moscow, Russia.
• Serge Vladut, Aix Marseille Universite, France, and Institute for Information Transmission Problems, Moscow, Russia.
• Dmitri Nogin, Institute for Information Transmission Problems, Moscow, Russia.