Algebraic Function Fields and Codes

1. Front Matter

   Pages I-XIII

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2. Foundations of the Theory of Algebraic Function Fields

   Pages 1-44

3. Algebraic Geometry Codes

   Pages 45-65

4. Extensions of Algebraic Function Fields

   Pages 67-154

5. Differentials of Algebraic Function Fields

   Pages 155-184

6. Algebraic Function Fields over Finite Constant Fields

   Pages 185-215

7. Examples of Algebraic Function Fields

   Pages 217-241

8. Asymptotic Bounds for the Number of Rational Places

   Pages 243-287

9. More about Algebraic Geometry Codes

   Pages 289-309

10. Subfield Subcodes and Trace Codes

    Pages 311-326

11. Back Matter

    Pages 327-359

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15 years after the ?rst printing of Algebraic Function Fields and Codes,the mathematics editors of Springer Verlag encouraged me to revise and extend the book. Besides numerous minor corrections and amendments, the second edition di?ers from the ?rst one in two respects. Firstly I have included a series of exercises at the end of each chapter. Some of these exercises are fairly easy and should help the reader to understand the basic concepts, others are more advanced and cover additional material. Secondly a new chapter titled “Asymptotic Bounds for the Number of Rational Places” has been added. This chapter contains a detailed presentation of the asymptotic theory of function ?elds over ?nite ?elds, including the explicit construction of some asymptotically good and optimal towers. Based on these towers, a complete and self-contained proof of the Tsfasman-Vladut-Zink Theorem is given. This theorem is perhaps the most beautiful application of function ?elds to coding theory. The codes which are constructed from algebraic function ?elds were ?rst introduced by V. D. Goppa. Accordingly I referred to them in the ?rst edition as geometric Goppa codes. Since this terminology has not generally been - cepted in the literature, I now use the more common term algebraic geometry codes or AG codes. I would like to thank Alp Bassa, Arnaldo Garcia, Cem Guneri, ¨ Sevan Harput and Alev Topuzo? glu for their help in preparing the second edition.

• Algebra
• Algebraische Funktionenkörper
• Codierungstheorie
• Funktionen
• algebraic curves
• algebraic function fields
• algebraic geometric codes
• coding theory
• computer science
• cryptography
• electrical engineering
• geometry
• information
• linear optimizatio
• information and communication, circuits

From the reviews of the second edition:

"In this book we have an exposition of the theory of function fields in one variable from the algebraic point of view … . The book is carefully written, the concepts are well motivated and plenty of examples help to understand the ideas and proofs and so it can be used as a textbook for an introductory course on the (classical) arithmetic of function fields with an application to coding theory." (Felipe Zaldivar, MAA Online, January, 2009)

• Sabanci University, Orhanli, Turkey

  Henning Stichtenoth