This introductory book, which grew out of lectures given at the Mathematics Institute of Würzburg University, proposes a combination of coding theory and number theory. Chapter 1 gives a standard course of linear codes. The next two chapters “Diophantine equations and cyclic codes”, “Elliptic curves, Hecke operators and weight distribution of codes” treat a link between coding theory and number theory. Chapter 4 “Algebraic-geometric codes and modular curve codes”, is a systematic study of algebraic-geometric codes and in Chapter 5 “Theta functions and self-dual codes” a connection between binary linear codes and theta functions is discussed. The appendix entitled “The Kloosterman codes and distribution of the weights”, contains essentially the contents of two papers published elsewhere where Kloosterman sums (or hyper-Kloosterman sums) play a prominent role.
This well-written book is designed to teach undergraduates and graduates the basic ideas and techniques of coding theory and number theory.