# Codes and Curves

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*Judy L. Walker*

When information is transmitted, errors are likely to occur. This
problem has become increasingly important as tremendous amounts of
information are transferred electronically every day. Coding theory
examines efficient ways of packaging data so that these errors can be
detected, or even corrected.

The traditional tools of coding theory have come from combinatorics
and group theory. Since the work of Goppa in the late 1970s, however,
coding theorists have added techniques from algebraic geometry to
their toolboxes. In particular, by re-interpreting the Reed-Solomon
codes as coming from evaluating functions associated to divisors on
the projective line, one can see how to define new codes based on
other divisors or on other algebraic curves. For instance, using
modular curves over finite fields, Tsfasman, Vladut, and Zink showed
that one can define a sequence of codes with asymptotically better
parameters than any previously known codes.

This book is based on a series of lectures the author gave as
part of the IAS/Park City Mathematics Institute (Utah) program on arithmetic algebraic geometry. Here,
the reader is introduced to the exciting field of algebraic geometric
coding theory. Presenting the material in the same conversational
tone of the lectures, the author covers linear codes, including cyclic
codes, and both bounds and asymptotic bounds on the parameters of
codes. Algebraic geometry is introduced, with particular attention
given to projective curves, rational functions and divisors. The
construction of algebraic geometric codes is given, and the
Tsfasman-Vladut-Zink result mentioned above is discussed.

No previous experience in coding theory or algebraic geometry is
required. Some familiarity with abstract algebra, in particular
finite fields, is assumed. However, this material is reviewed in two
appendices. There is also an appendix containing projects that
explore other codes not covered in the main text.

This book is published in cooperation with IAS/Park City Mathematics Institute

#### Readership

Undergraduates in mathematics; mathematicians interested in coding theory or algebraic geometry and the connections between the two subjects.

#### Reviews & Endorsements

A great addition to an abstract algebra course or an algebra topics course. Written in a conversational tone; packed with accessible examples and theory.

-- MAA Monthly

A useful supplement to an undergraduate text in coding theory to give students a basic introduction to algebraic geometry codes.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Codes and Curves

- Cover Cover11 free
- Other titles in this series i2 free
- Title page v6 free
- Contents vii8 free
- Subseries preface ix10 free
- Preface xi12 free
- Introduction to coding theory 114 free
- Bounds on codes 922 free
- Algebraic curves 1730
- Nonsingularity and the genus 2538
- Points, functions, and divisors on curves 3144
- Algebraic geometry codes 3952
- Good codes from algebraic geometry 4356
- Abstract algebra review 4760
- Finite fields 5770
- Projects 6376
- Bibliography 6780
- Back Cover Back Cover182