Softcover ISBN:  9780821826287 
Product Code:  STML/7 
List Price:  $49.00 
Individual Price:  $39.20 
eBook ISBN:  9781470418236 
Product Code:  STML/7.E 
List Price:  $39.00 
Individual Price:  $31.20 
Softcover ISBN:  9780821826287 
eBook: ISBN:  9781470418236 
Product Code:  STML/7.B 
List Price:  $88.00 $68.50 
Softcover ISBN:  9780821826287 
Product Code:  STML/7 
List Price:  $49.00 
Individual Price:  $39.20 
eBook ISBN:  9781470418236 
Product Code:  STML/7.E 
List Price:  $39.00 
Individual Price:  $31.20 
Softcover ISBN:  9780821826287 
eBook ISBN:  9781470418236 
Product Code:  STML/7.B 
List Price:  $88.00 $68.50 

Book DetailsStudent Mathematical LibraryIAS/Park City Mathematics SubseriesVolume: 7; 2000; 66 ppMSC: Primary 11; 94; Secondary 14
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 reinterpreting the ReedSolomon 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 TsfasmanVladutZink 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.ReadershipUndergraduates in mathematics; mathematicians interested in coding theory or algebraic geometry and the connections between the two subjects.

Table of Contents

Chapters

Chapter 1. Introduction to coding theory

Chapter 2. Bounds on codes

Chapter 3. Algebraic curves

Chapter 4. Nonsingularity and the genus

Chapter 5. Points, functions, and divisors on curves

Chapter 6. Algebraic geometry codes

Chapter 7. Good codes from algebraic geometry

Appendix A. Abstract algebra review

Appendix B. Finite fields

Appendix C. Projects


Additional Material

Reviews

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


RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
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 reinterpreting the ReedSolomon 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 TsfasmanVladutZink 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.
Undergraduates in mathematics; mathematicians interested in coding theory or algebraic geometry and the connections between the two subjects.

Chapters

Chapter 1. Introduction to coding theory

Chapter 2. Bounds on codes

Chapter 3. Algebraic curves

Chapter 4. Nonsingularity and the genus

Chapter 5. Points, functions, and divisors on curves

Chapter 6. Algebraic geometry codes

Chapter 7. Good codes from algebraic geometry

Appendix A. Abstract algebra review

Appendix B. Finite fields

Appendix C. Projects

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