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.

This book links two subjects: algebraic geometry and coding theory. It uses a novel approach based on the theory of algebraic function fields. Coverage includes the Riemann-Rock theorem, zeta functions and Hasse-Weil's theorem as well as Goppa' s algebraic-geometric codes and other traditional codes. It will be useful to researchers in algebraic geometry and coding theory and computer scientists and engineers in information transmission.

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.

The theory of algebraic function fields over finite fields has its origins in number theory. However, after Goppas discovery of algebraic geometry codes around 1980, many applications of function fields were found in different areas of mathematics and information theory. This book presents survey articles on some of these new developments. The topics focus on material which has not yet been presented in other books or survey articles.

This book links two subjects: algebraic geometry and coding theory. It uses a novel approach based on the theory of algebraic function fields. Coverage includes the Riemann-Rock theorem, zeta functions and Hasse-Weil's theorem as well as Goppa' s algebraic-geometric codes and other traditional codes. It will be useful to researchers in algebraic geometry and coding theory and computer scientists and engineers in information transmission.

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.

Leading researchers in the field of coding theory and cryptography present their newest findings, published here for the first time following a presentation at the International Conference on Coding Theory, Cryptography and Related Areas. The authors include Tom Hoeholdt, Henning Stichtenoth, and Horacio Tapia-Recillas.

About ten years ago, V.D. Goppa found a surprising connection between the theory of algebraic curves over a finite field and error-correcting codes. The aim of the meeting "Algebraic Geometry and Coding Theory" was to give a survey on the present state of research in this field and related topics. The proceedings contain research papers on several aspects of the theory, among them: Codes constructed from special curves and from higher-dimensional varieties, Decoding of algebraic geometric codes, Trace codes, Exponen- tial sums, Fast multiplication in finite fields, Asymptotic number of points on algebraic curves, Sphere packings.