Henning Stichtenoth - Böcker

The theory of algebraic function fields over finite fields has its origins in number theory. However, after Goppa`s discovery of algebraic geometry codes around 1980, many applications of function fields were found in different areas of mathematics and information theory, such as coding theory, sphere packings and lattices, sequence design, and cryptography. The use of function fields often led to better results than those of classical approaches.This book presents survey articles on some of these new developments. Most of the material is directly related to the interaction between function fields and their various applications; in particular the structure and the number of rational places of function fields are of great significance. The topics focus on material which has not yet been presented in other books or survey articles. Wherever applications are pointed out, a special effort has been made to present some background concerning their use.

Thisvolumerepresentstherefereedproceedingsofthe7thInternationalC- ference on Finite Fields and Applications (F 7) held during May 5-9, q 2003, in Toulouse, France. The conference was hosted by the Pierre Baudis C- gress Center, downtown, and held at the excellent conference facility. This event continued a series of biennial international conferences on Finite Fields and - plications, following earlier meetings at the University of Nevada at Las Vegas (USA) in August 1991 and August 1993, the University of Glasgow (UK) in July 1995, the University of Waterloo (Canada) in August 1997, the Univ- sity of Augsburg (Germany) in August 1999, and the Universidad Aut' onoma Metropolitana-Iztapalapa, in Oaxaca (Mexico) in 2001. The Organizing Committee of F 7 consisted of Claude Carlet (INRIA, Paris, q France), Dieter Jungnickel (University of Augsburg, Germany), Gary Mullen (Pennsylvania State University, USA), Harald Niederreiter (National University of Singapore, Singapore), Alain Poli, Chair (Paul Sabatier University, Toulouse, France), Henning Stichtenoth (Essen University, Germany), and Horacio Tapia- Recillas (Universidad Aut' onoma Metropolitan-Iztapalapa, Mexico).The program of the conference consisted of four full days and one half day of sessions, with eight invited plenary talks, and close to 60 contributed talks.

About ten years ago, V.D. Goppa found a surprisingconnection between the theory of algebraic curves over afinite field and error-correcting codes. The aim of themeeting "Algebraic Geometry and Coding Theory" was to give asurvey on the present state of research in this field andrelated topics. The proceedings contain research papers onseveral aspects of the theory, among them: Codes constructedfrom special curves and from higher-dimensional varieties,Decoding of algebraic geometric codes, Trace codes, Exponen-tial sums, Fast multiplication in finite fields, Asymptoticnumber of points on algebraic curves, Sphere packings.

This book contains 23 contributions presented at the "International Conference on Coding Theory, Cryptography and Related Areas (ICCC)", held in Guanajuato, Mexico, in April 1998.It comprises a series of research papers on various aspects of coding theory (geometric-algebraic, decoding, exponential sums, etc.) and cryptography (discrete logarithm problem, public key cryptosystems, primitives, etc.), as well as in other research areas, such as codes over finite rings and some aspects of function fields and algebraic geometry over finite fields.The book contains new results on the subject, never published in any other form. It will be useful to students, researchers, professionals, and tutors interested in this area of research.

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.

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.

This volume represents the refereed proceedings of the "Sixth International Conference on Finite Fields and Applications (Fq6)" held in the city of Oaxaca, Mexico, between 22-26 May 200l. The conference was hosted by the Departmento do Matermiticas of the U niversidad Aut6noma Metropolitana- Iztapalapa, Nlexico. This event continued a series of biennial international conferences on Finite Fields and Applications, following earlier meetings at the University of Nevada at Las Vegas (USA) in August 1991 and August 1993, the University of Glasgow (Scotland) in July 1995, the University of Waterloo (Canada) in August 1997, and at the University of Augsburg (Ger- many) in August 1999. The Organizing Committee of Fq6 consisted of Dieter Jungnickel (University of Augsburg, Germany), Neal Koblitz (University of Washington, USA), Alfred }. lenezes (University of Waterloo, Canada), Gary Mullen (The Pennsylvania State University, USA), Harald Niederreiter (Na- tional University of Singapore, Singapore), Vera Pless (University of Illinois, USA), Carlos Renteria (lPN, Mexico). Henning Stichtenoth (Essen Univer- sity, Germany). and Horacia Tapia-Recillas, Chair (Universidad Aut6noma l'vIetropolitan-Iztapalapa.Mexico). The program of the conference consisted of four full days and one half day of sessions, with 7 invited plenary talks, close to 60 contributed talks, basic courses in finite fields. cryptography and coding theory and a series of lectures at local educational institutions. Finite fields have an inherently fascinating structure and they are im- portant tools in discrete mathematics.