This book grew out of our lectures given in the Oberseminar on 'Cod- ing
Theory and Number Theory' at the Mathematics Institute of the Wiirzburg
University in the Summer Semester, 2001. The coding the- ory combines
mathematical elegance and some engineering problems to an unusual
degree. The major advantage of studying coding theory is the beauty of
this particular combination of mathematics and engineering. In this book
we wish to introduce some practical problems to the math- ematician and
to address these as an essential part of the development of modern
number theory. The book consists of five chapters and an appendix.
Chapter 1 may mostly be dropped from an introductory course of linear
codes. In Chap- ter 2 we discuss some relations between the number of
solutions of a diagonal equation over finite fields and the weight
distribution of cyclic codes. Chapter 3 begins by reviewing some basic
facts from elliptic curves over finite fields and modular forms, and
shows that the weight distribution of the Melas codes is represented by
means of the trace of the Hecke operators acting on the space of cusp
forms. Chapter 4 is a systematic study of the algebraic-geometric codes.
For a long time, the study of algebraic curves over finite fields was
the province of pure mathematicians. In the period 1977 - 1982, V. D.
Goppa discovered an amazing connection between the theory of algebraic
curves over fi- nite fields and the theory of q-ary codes.
