This introductory book, which grew out of lectures given at the Mathematics Institute of W rzburg University, proposes a combination of coding theory and number theory. Chapter 1 gives a standard course of linear codes. The next two chapters treat a link between coding theory and number theory. Chapter 4 is a systematic study of algebraic-geometric codes and in Chapter 5 a connection between binary linear codes and theta functions is discussed. The book is designed to teach undergraduates and graduates the basic ideas and techniques of coding theory and number theory.
Table of Contents
1. Linear Codes.- 2. Diophantine Equations and Cyclic Codes.- 3. Elliptic Curves, Hecke Operators and Weight Distribution of Codes.- 4. Algebraic-Geometric Codes and Modular Curve Codes.- 5. Theta Functions and Self-Dual Codes.- The Kloosterman Codes and Distribution of the Weights.- 1 Introduction.- 2 Melas code and Kloosterman sums.- 3 Hyper-Kloosterman code.- 4 Quasi-cyclic property.- 5 Weight distribution.- 7 A divisibility theorem for Hamming weights.- References.