Review of algebraic function fields and codes by Henning Stichtenoth
The book being reviewed, “Algebraic Function Fields and Codes” by Henning Stichtenoth, gives a rigorous, systematic and thorough treatment of the theory of algebraic function fields and its applications in coding theory. Algebraic function fields are natural objects that arise in the study of algebraic curves and exponential sums over finite fields, and have found widespread applications in the areas of coding theory and pseudorandomness. This is the second edition of a book that is already one of the standard references in the area of algebraicgeometry codes. The new edition features a new chapter on curves with many rational points, as well as a large number of exercises at the end of every chapter. The new chapter is a particularly welcome addition, since the book now contains an explicit construction and complete analysis of excellent error-correcting codes that beat random codes (which is one of the most celebrated applications of the theory of algebraic function fields to coding theory and theoretical computer science). Below we will give a short introduction to algebraic function fields and a brief description of one of their major applications in coding theory We will then proceed to discuss the contents of this book.