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Algebraic Function Fields and Codes

Henning Stichtenoth (Autor)

Buch | Hardcover

XIV, 360 Seiten

2008 | 2nd ed. 2008
Springer Berlin (Verlag)
978-3-540-76877-7 (ISBN)

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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.

Foundations of the Theory of Algebraic Function Fields.- Algebraic Geometry Codes.- Extensions of Algebraic Function Fields.- Differentials of Algebraic Function Fields.- Algebraic Function Fields over Finite Constant Fields.- Examples of Algebraic Function Fields.- Asymptotic Bounds for the Number of Rational Places.- More about Algebraic Geometry Codes.- Subfield Subcodes and Trace Codes.

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)

Erscheint lt. Verlag	20.11.2008
Reihe/Serie	Graduate Texts in Mathematics
Zusatzinfo	XIV, 360 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	676 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Schlagworte	Algebra • algebraic curves • Algebraic Function Fields • algebraic geometric codes • Algebraische Funktionenkörper • Algebraische Geometrie, insb. algebraische Kurven • Codierung • Codierung / Kodierung • Codierungstheorie • coding theory • Computer Science • cryptography • Electrical Engineering • Funktionen • Geometry • Information • Information and Communication, Circuits • linear optimizatio
ISBN-10	3-540-76877-7 / 3540768777
ISBN-13	978-3-540-76877-7 / 9783540768777
Zustand	Neuware