Chapter 1
Introduction to Coding
Theory
1.1. Overview
Whenever data is transmitted across a channel, errors are likely to
occur. It is the goal of coding theory to find efficient ways of encod-
ing the data so that these errors can be detected, or even corrected.
Traditionally, the main tools used in coding theory have been those
of combinatorics and group theory. In 1977, V. D. Goppa defined
algebraic geometric codes [Go], thus allowing a wide range of tech-
niques from algebraic geometry to be applied. Goppa’s idea has had
a great impact on the field. Not long after Goppa’s original paper,
Tsfasman, Vladut and Zink [TVZ] used modular curves to construct
a sequence of codes with asymptotically better parameters than any
previously known codes. The goal of this course is to introduce you to
some of the basics of coding theory, algebraic geometry, and algebraic
geometric codes.
Before we write down a rigorous definition of a code, let’s look
at some examples. Probably the most commonly seen code in day-
to-day life is the International Standardized Book Number (ISBN)
Code. Every book is assigned an ISBN, and that ISBN is typically
displayed on the back cover of the book. For example, the ISBN for
The Theory of Error-Correcting Codes by MacWilliams and Sloane
1
http://dx.doi.org/10.1090/stml/007/01
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