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By Yves Aubry, Gilles Lachaud

Résumé :
Arithmétique, géométrie et théorie des codes (AGCT 2003)
En mai 2003 se sont tenus au Centre foreign de Rencontres Mathématiques à Marseille (France), deux événements centrés sur l'Arithmétique, los angeles Géométrie et leurs purposes à l. a. théorie des Codes ainsi qu'à l. a. Cryptographie : une école Européenne ``Géométrie Algébrique et Théorie de l'Information'' ainsi que los angeles 9ème édition du colloque overseas ``Arithmétique, Géométrie et Théorie des Codes''. Certains des cours et des conférences font l'objet d'un article publié dans ce quantity. Les thèmes abordés furent à l. a. fois théoriques pour certains et tournés vers des functions pour d'autres : variétés abéliennes, corps de fonctions et courbes sur les corps finis, groupes de Galois de pro-p-extensions, fonctions zêta de Dedekind de corps de nombres, semi-groupes numériques, nombres de Waring, complexité bilinéaire de los angeles multiplication dans les corps finis et problèmes de nombre de classes.

Mots clefs : Fonctions zêta, variétés abéliennes, corps de fonctions, courbes sur les corps finis, excursions de corps de fonctions, corps finis, graphes, semi-groupes numériques, polynômes sur les corps finis, cryptographie, courbes hyperelliptiques, représentations p-adiques, excursions de corps de classe, groupe de Galois, issues rationels, fractions keeps, régulateurs, nombre de periods d'idéaux, complexité bilinéaire, jacobienne hyperelliptiques

Abstract:
In might 2003, occasions were held within the ``Centre foreign de Rencontres Mathématiques'' in Marseille (France), dedicated to mathematics, Geometry and their purposes in Coding concept and Cryptography: an ecu institution ``Algebraic Geometry and knowledge Theory'' and the 9-th foreign convention ``Arithmetic, Geometry and Coding Theory''. a number of the classes and the meetings are released during this quantity. the themes have been theoretical for a few ones and became in the direction of functions for others: abelian forms, functionality fields and curves over finite fields, Galois staff of pro-p-extensions, Dedekind zeta features of quantity fields, numerical semigroups, Waring numbers, bilinear complexity of the multiplication in finite fields and sophistication quantity problems.

Key phrases: Zeta services, abelian types, capabilities fields, curves over finite fields, towers of functionality fields, finite fields, graphs, numerical semigroups, polynomials over finite fields, cryptography, hyperelliptic curves, p-adic representations, type box towers, Galois teams, rational issues, persevered fractions, regulators, perfect category quantity, bilinear complexity, hyperelliptic jacobians

Class. math. : 14H05, 14G05, 11G20, 20M99, 94B27, 11T06, 11T71, 11R37, 14G10, 14G15, 11R58, 11A55, 11R42, 11Yxx, 12E20, 14H40, 14K05

Table of Contents

* P. Beelen, A. Garcia, and H. Stichtenoth -- On towers of functionality fields over finite fields
* M. Bras-Amorós -- Addition habit of a numerical semigroup
* O. Moreno and F. N. Castro -- at the calculation and estimation of Waring quantity for finite fields
* G. Frey and T. Lange -- Mathematical history of Public Key Cryptography
* A. Garcia -- On curves over finite fields
* F. Hajir -- Tame pro-p Galois teams: A survey of contemporary work
* E. W. Howe, okay. E. Lauter, and J. most sensible -- unnecessary curves of genus 3 and four
* D. Le Brigand -- actual quadratic extensions of the rational functionality box in attribute two
* S. R. Louboutin -- particular higher bounds for the residues at s=1 of the Dedekind zeta services of a few completely actual quantity fields
* S. Ballet and R. Rolland -- at the bilindar complexity of the multiplication in finite fields
* Yu. G. Zarhin -- Homomorphisms of abelian types

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Sample text

Mattson used the Delsarte’s bound, and the final paper by Helleseth invokes the Weil-Carlitz-Uchiyama bound. 11 in [10]. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 2005 O. N. 3. — Let f = 2t + 1 and α be a primitive root of F2f . The code C with zeroes α, αd , αd , where d = 2t−1 + 1, and d = 2t + 1 has covering radius 5. Let d1 , d2 be distinct natural numbers. 1. 4. — Let α be a primitive root of F2f and let C be the code of length n = 2f − 1 with zeros αd1 , αd2 over F2f . We assume that the minimum distance of C is 5.

T11r + t22r + · · · + tnNr r ≡ 0 mod q − 1, d111 t111 + d121 t121 + · · · + d1Nr r t1Nr r ≡ 0 mod q − 1, d211 t211 + d221 t221 + · · · + d2Nr r t2Nr r ≡ 0 mod q − 1, .. dn11 tn11 + dn21 tn21 + · · · + dnNr r tnNr r ≡ 0 mod q − 1. Now we are ready to state the main theorem of [15]. 1. — Let G be the following class of polynomials G = {a11 X d11 + · · · + a1N1 X d1N1 , · · · , ar1 X dr1 + · · · , arNr X drNr | aij ∈ Fq }. With L as above, there are polynomials F1 , . . , Fr in G, such that |N | is divisible by pL−f r but not divisible by pL+1−f r .

Vol. 2643, Springer, Berlin, 2003, p. 204–215. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 2005 28 ´ M. D. Thesis, Universitat Polit`ecnica [3] M. Bras-Amoro de Catalunya, Barcelona, 2003. ´ s – Acute semigroups, the order bound on the minimum distance, and [4] M. Bras-Amoro the Feng-Rao improvements, IEEE Trans. Inform. Theory 50 (2004), no. 6, p. 1282– 1289. E. O’Sullivan – The correction capability of the Berlekamp[5] M. Bras-Amoro Massey-Sakata algorithm with majority voting, submitted, 2004. ´ n & C.

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