Publications HAL de yves,Aubry

2024

Pré-publication, Document de travail

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Yves Aubry, Fabien Herbaut, Ali Issa. TRINOMIALS WITH HIGH DIFFERENTIAL UNIFORMITY. 2024. ⟨hal-04546074⟩
resume
The context of this work is the study of the differential uniformity of polynomials defined over finite fields of even characteristic. We provide infinite families of trinomials with high differential uniformity when the base field is large enough. It means in particular that these trinomials are not exceptional almost perfect nonlinear.
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https://hal.science/hal-04546074/file/Aubry_Herbaut_Issa_Trinomials_HAL.pdf BibTex
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Yves Aubry, Marc Perret. MAXIMUM NUMBER OF RATIONAL POINTS ON HYPERSURFACES IN WEIGHTED PROJECTIVE SPACES OVER FINITE FIELDS. 2024. ⟨hal-04449392⟩
resume
An upper bound for the maximum number of rational points on an hypersurface in a projective space over a finite field has been conjectured by Tsfasman and proved by Serre in 1989. The analogue question for hypersurfaces on weighted projective spaces has been considered by Castryck, Ghorpade, Lachaud, O'Sullivan, Ram and the first author in 2017. A conjecture has been proposed there and proved in the particular case of the dimension 2. We prove here the conjecture in any dimension provided the second weight is also equal to one.
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https://hal.science/hal-04449392/file/Aubry_Perret_Weighted_Projective_Spaces.pdf BibTex

2023

Article dans une revue

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Yves Aubry, Ali Issa, Fabien Herbaut. Polynomials with maximal differential uniformity and the exceptional APN conjecture. Journal of Algebra, 2023, 635, pp.822-837. ⟨10.1016/j.jalgebra.2023.07.017⟩. ⟨hal-03739111⟩
resume
We contribute to the exceptional APN conjecture by showing that no polynomial of degree m = 2 r (2 ℓ + 1) where gcd(r, ℓ) 2, r 2, ℓ 1 with a nonzero second leading coefficient can be APN over infinitely many extensions of the base field. More precisely, we prove that for n sufficiently large, all polynomials of F 2 n [x] of such a degree with a nonzero second leading coefficient have a differential uniformity equal to m − 2.
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Article dans une revue
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https://hal.science/hal-03739111/file/Aubry_Herbaut_Issa_220726.pdf BibTex

Proceedings/Recueil des communications

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Yves Aubry, Fabien Herbaut, Julien Monaldi. CLOSED POINTS ON CURVES OVER FINITE FIELDS. 2023. ⟨hal-04245190⟩
resume
We are interested in the quantity ρ(q, g) defined as the smallest positive integer such that r ≥ ρ(q, g) implies that any absolutely irreducible smooth projective algebraic curve defined over F q of genus g has a closed point of degree r. We provide general upper bounds for this number and its exact value for g = 1, 2 and 3. We also improve the known upper bounds on the number of closed points of degree 2 on a curve.
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Proceedings/Recueil des communications
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https://hal.science/hal-04245190/file/Rho%28q%2Cg%29_Aubry_Herbaut_Monaldi.pdf BibTex

2021

Article dans une revue

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Yves Aubry, Elena Berardini, Fabien Herbaut, Marc Perret. Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus. Finite Fields and Their Applications, 2021, ⟨10.1016/j.ffa.2020.101791⟩. ⟨hal-02100210v2⟩
resume
We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain low genus curves. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization.
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https://hal.science/hal-02100210/file/ABHP_Algebraic_geometry_codes_over_abelian_surfaces_v3.pdf BibTex
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Yves Aubry, Elena Berardini, Fabien Herbaut, Marc Perret. Bounds on the minimum distance of algebraic geometry codes defined over some families of surfaces. Contemporary mathematics, 2021, 770. ⟨hal-02411489v3⟩
resume
We prove lower bounds for the minimum distance of algebraic geometry codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds for surfaces whose arithmetic Picard number equals one, surfaces without curves with small self-intersection and fibered surfaces. Finally we specify our bounds to the case of surfaces of degree $d\geq 3$ embedded in $\mathbb{P}^3$.
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Article dans une revue
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https://hal.science/hal-02411489/file/Aubry_Berardini_Herbaut_Perret_v3.pdf BibTex

Ouvrages

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Yves Aubry, Pierre Barthélémy, Nadia El Mrabet. Special issue from mathematics to embedded devices. 13 (4), pp.475-477, 2021, ⟨10.1007/s12095-021-00502-1⟩. ⟨hal-03424622⟩
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Ouvrages
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https://hal.science/hal-03424622/file/Aubry2021_Article_SpecialIssueFromMathematicsToE.pdf BibTex

2019

Article dans une revue

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Yves Aubry, Fabien Herbaut, José Felipe Voloch. Maximal Differential Uniformity Polynomials. Acta Arithmetica, 2019, 188 (4), pp.345-366. ⟨10.4064/aa170806-11-7⟩. ⟨hal-01823955⟩
resume
We provide explicit infinite families of integers m such that all the polynomials of F2n [x] of degree m have maximal differential uniformity for n large enough. We also prove a conjecture of the third author for these families.
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https://hal.science/hal-01823955/file/Aubry_Herbaut_Voloch_Differential_Uniformity_Revised.pdf BibTex

Ouvrages

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Yves Aubry, Everett W. Howe, Christophe Ritzenthaler. Arithmetic Geometry: Computation and Applications. 2019, ⟨10.1090/conm/722⟩. ⟨hal-02188519⟩
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Poster de conférence

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Florence Niemetzky, Basma Guarmit, Yves Aubry, Nathan Chapuis, Jean-François Carod, et al.. État des lieux de la population diabétique de la commune de Grand Santi en 2017. 3ème journée des travaux scientifiques des jeunes médecins de Guyane, Dec 2019, Cayenne, Guyane française. ⟨hal-03670399⟩
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Poster de conférence
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https://hal.univ-guyane.fr/hal-03670399/file/posterFlo%281%29.pdf BibTex

2018

Article dans une revue

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Yves Aubry, Fabien Herbaut. Differential uniformity and second order derivatives for generic polynomials. Journal of Pure and Applied Algebra, 2018, 222 (5), pp.1095-1110. ⟨10.1016/j.jpaa.2017.06.009⟩. ⟨hal-01266567⟩
resume
For any polynomial $f$ of ${\mathbb F}_{2^n}[x]$ we introduce the following characteristic of the distribution of its second order derivative, which extends the differential uniformity notion: $$\delta^2(f):=\max_{\substack{ \alpha \in {\mathbb F}_{2^n}^{\ast} ,\alpha' \in {\mathbb F}_{2^n}^{\ast} ,\beta \in {\mathbb F}_{2^n} \\ \alpha\not=\alpha'}} \sharp\{x\in{\mathbb F}_{2^n} \mid D_{\alpha,\alpha'}^2f(x)=\beta\}$$ where $D_{\alpha,\alpha'}^2f(x):=D_{\alpha'}(D_{\alpha}f(x))=f(x)+f(x+\alpha)+f(x+\alpha')+f(x+\alpha+\alpha')$ is the second order derivative. Our purpose is to prove a density theorem relative to this quantity, which is an analogue of a density theorem proved by Voloch for the differential uniformity.
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https://hal.science/hal-01266567/file/Aubry_Herbaut_170320.pdf BibTex

2017

Article dans une revue

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Yves Aubry, Dimitrios Poulakis. Thue equations and CM-fields. Ramanujan Journal, 2017, 42 (1), pp.145 - 156. ⟨10.1007/s11139-015-9749-x⟩. ⟨hal-01044876v5⟩
resume
We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations $F(X,Y) = b$ defined over a totally real number field $K$, assuming that $F(X,1)$ has a root $\alpha$ such that $K(\alpha)$ is a CM-field. Furthermore, we give an algorithm for the computation of the integral solutions of such an equation.
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https://hal.science/hal-01044876/file/Aubry_Poulakis_v5.pdf BibTex
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Yves Aubry, Annamaria Iezzi. Optimal and maximal singular curves. Contemporary mathematics, 2017, Arithmetic, Geometry and Coding Theory, 686, pp.31--43. ⟨10.1090/conm/686/13776⟩. ⟨hal-01212624⟩
resume
Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field $\mathbb F_q$. This bound enables us to provide explicit conditions on $q, g$ and $\pi$ for the non-existence of absolutely irreducible projective algebraic curves defined over $\mathbb F_q$ of geometric genus $g$, arithmetic genus $\pi$ and with $N_q(g)+\pi-g$ rational points. Moreover, for $q$ a square, we study the set of pairs $(g,\pi)$ for which there exists a maximal absolutely irreducible projective algebraic curve defined over $\mathbb F_q$ of geometric genus $g$ and arithmetic genus $\pi$, i.e. with $q+1+2g\sqrt{q}+\pi-g$ rational points.
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https://hal.science/hal-01212624/file/Aubry_Iezzi.pdf BibTex

Chapitre d'ouvrage

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Yves Aubry, Wouter Castryck, Sudhir R Ghorpade, Gilles Lachaud, Michael E O 'Sullivan, et al.. Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory. Howe E., Lauter K., Walker J. Algebraic Geometry for Coding Theory and Cryptography, Association for Women in Mathematics Series, volume 9, Springer, Cham, pp.25-61, 2017, 978-3-319-63931-4. ⟨10.1007/978-3-319-63931-4_2⟩. ⟨hal-01478729v2⟩
resume
We consider the question of determining the maximum number of Fq-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field Fq, or in other words, the maximum number of zeros that a weighted homogeneous polynomial of a given degree can have in the corresponding weighted projective space over Fq. In the case of classical projective spaces, this question has been answered by J.-P. Serre. In the case of weighted projective spaces, we give some conjectures and partial results. Applications to coding theory are included and an appendix providing a brief compendium of results about weighted projective spaces is also included.
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Chapitre d'ouvrage
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https://amu.hal.science/hal-01478729/file/Hypersurfaces_in_weighted_projective_spaces-v2.pdf BibTex

2016

Article dans une revue

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Yves Aubry, Jean-Christophe Godin, Olivier Togni. Free choosability of outerplanar graphs. Graphs and Combinatorics, 2016, 32 (3), pp.851-859. ⟨10.1007/s00373-015-1625-3⟩. ⟨hal-00957298⟩
resume
A graph $G$ is free $(a,b)$-choosable if for any vertex $v$ with $b$ colors assigned and for any list of colors of size $a$ associated with each vertex $u\ne v$, the coloring can be completed by choosing for $u$ a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. In this note, a necessary and sufficient condition for a cycle to be free $(a,b)$-choosable is given. As a corollary, some choosability results are derived for graphs in which cycles are connected by a tree structure.
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Article dans une revue
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https://hal.science/hal-00957298/file/freechooscycle.pdf BibTex
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Yves Aubry, Safia Haloui. On the number of rational points on Prym varieties over finite fields. Glasgow Mathematical Journal, 2016, 58 (Issue 1), pp.55--68. ⟨10.1017/S0017089515000063⟩. ⟨hal-00843686⟩
resume
We give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2.
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Article dans une revue
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https://hal.science/hal-00843686/file/Aubry_Haloui.pdf BibTex

2015

Article dans une revue

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Yves Aubry, Annamaria Iezzi. On the maximum number of rational points on singular curves over finite fields. Moscow Mathematical Journal, 2015, 15 (4), pp.615--627. ⟨10.17323/1609-4514-2015-15-4-615-627⟩. ⟨hal-01103802v2⟩
resume
We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus π.
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https://hal.science/hal-01103802/file/20150817a_Singular_Aubry_Iezzi_1er_Oct_2015.pdf BibTex
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Yves Aubry, Daniel J. Katz, Philippe Langevin. Cyclotomy of Weil Sums of Binomials. Journal of Number Theory, 2015, 154, pp.160--178. ⟨10.1016/j.jnt.2015.02.011⟩. ⟨hal-00978918⟩
resume
The Weil sum $W_{K,d}(a)=\sum_{x \in K} \psi(x^d + a x)$ where $K$ is a finite field, $\psi$ is an additive character of $K$, $d$ is coprime to $|K^\times|$, and $a \in K^\times$ arises often in number-theoretic calculations, and in applications to finite geometry, cryptography, digital sequence design, and coding theory. Researchers are especially interested in the case where $W_{K,d}(a)$ assumes three distinct values as $a$ runs through $K^\times$. A Galois-theoretic approach is used here to prove a variety of new results that constrain which fields $K$ and exponents $d$ support three-valued Weil sums, and restrict the values that such Weil sums may assume.
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https://arxiv.org/pdf/1312.3889 BibTex

2014

Article dans une revue

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Yves Aubry, Daniel Katz, Philippe Langevin. Cyclotomie des sommes de Weil binomiales. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2014, 352 (5), pp.373-376. ⟨10.1016/j.crma.2014.03.001⟩. ⟨hal-00993189⟩
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Yves Aubry, Jean-Christophe Godin, Olivier Togni. Every triangle-free induced subgraph of the triangular lattice is $(5m,2m)$-choosable. Discrete Applied Mathematics, 2014, 166, pp.51--58. ⟨10.1016/j.dam.2013.09.028⟩. ⟨hal-00631398⟩
resume
A graph $G$ is $(a,b)$-choosable if for any color list of size $a$ associated with each vertex, one can choose a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. This paper proves that for any integer $m\ge 1$, every finite triangle-free induced subgraph of the triangular lattice is $(5m,2m)$-choosable.
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https://hal.science/hal-00631398/file/jcyo52choosV3.pdf BibTex

2013

Article dans une revue

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Yves Aubry, Philippe Langevin. On a conjecture of Helleseth. Lecture Notes in Computer Science, 2013, 8080, pp.113--118. ⟨10.1007/978-3-642-40663-8_12⟩. ⟨hal-00769143⟩
resume
We are concern about a conjecture proposed in the middle of the seventies by Hellesseth in the framework of maximal sequences and theirs cross-correlations. The conjecture claims the existence of a zero outphase Fourier coefficient. We give some divisibility properties in this direction.
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https://hal.science/hal-00769143/file/Aubry_Langevin.pdf BibTex
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Yves Aubry, Safia Haloui, Gilles Lachaud. On the number of points on abelian and Jacobian varieties over finite fields. Acta Arithmetica, 2013, 160 (3), pp.201--241. ⟨10.4064/aa160-3-1⟩. ⟨hal-00978916⟩
resume
This article has roughly a threefold aim. The first is to provide a series of upper and lower bounds for the number of points on an abelian variety defined over a finite field. Our second aim is to obtain specific lower bounds in the special case where the abelian variety is the Jacobian of a smooth, projective, absolutely irreducible algebraic curve. The third aim is to give exact values for the maximum and the minimum number of rational points on Jacobian varieties of dimension 2.
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https://hal.science/hal-00978916/file/Aubry_Haloui_Lachaud.Acta_Arithmetica_2013.pdf BibTex

Communication dans un congrès

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Yves Aubry, Philippe Langevin. On a Conjecture of Helleseth. CAI, 2013, Porquerolles, France. ⟨hal-00993192⟩
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2012

Article dans une revue

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Yves Aubry, Safia Haloui, Gilles Lachaud. Sur le nombre de points rationnels des variétés abéliennes et des Jacobiennes sur les corps finis. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2012, 350 (19-20), pp.907--910. ⟨10.1016/j.crma.2012.10.001⟩. ⟨hal-00978914⟩
resume
Nous établissons de nouvelles majorations et minorations pour le nombre de points rationnels des variétés abéliennes et des Jacobiennes sur un corps fini. Nous déterminons de plus les nombres maximum et minimum de points rationnels des surfaces Jacobiennes sur un corps fini donné. We give upper and lower bounds for the number of points on abelian and Jacobian varieties over finite fields. We also determine the values for the maximum and minimum number of points on Jacobian surfaces on a given finite field.
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https://hal.science/hal-00978914/file/AHL_Cras_2.pdf BibTex
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Yves Aubry, Jean-Christophe Godin, Olivier Togni. Vectorial solutions to list multicoloring problems on graphs. Advances and Applications in Discrete Mathematics, 2012, 9 (2), pp.65 --81. ⟨hal-00672373⟩
resume
For a graph $G$ with a given list assignment $L$ on the vertices, we give an algebraical description of the set of all weights $w$ such that $G$ is $(L,w)$-colorable, called permissible weights. Moreover, for a graph $G$ with a given list $L$ and a given permissible weight $w$, we describe the set of all $(L,w)$-colorings of $G$. By the way, we solve the {\sl channel assignment problem}. Furthermore, we describe the set of solutions to the {\sl on call problem}: when $w$ is not a permissible weight, we find all the nearest permissible weights $w'$. Finally, we give a solution to the non-recoloring problem keeping a given subcoloring.
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https://hal.science/hal-00672373/file/ChannelAssignmentPaper.pdf BibTex

Ouvrages

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Yves Aubry, Christophe Ritzenthaler, Alexey Zykin. Arithmetic, Geometry, Cryptography and Coding Theory 2011. American Mathematical Society, pp.200, 2012, ⟨10.1090/conm/574⟩. ⟨hal-00979715⟩
resume
Proceedings of AGCT-2011 and GeoCrypt-2011
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Pré-publication, Document de travail

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Yves Aubry, Safia Haloui, Gilles Lachaud. Number of points on abelian and Jacobian varieties over finite fields. 2012. ⟨hal-00662352v3⟩
resume
We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.
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https://hal.science/hal-00662352/file/Abelian.pdf BibTex

2011

Ouvrages

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Yves Aubry, Claude Carlet, Philippe Langevin, Pascal Véron. Guest editorial for the special issue for Jacques Wolfmann. Editors: Yves Aubry and Claude Carlet and Philippe Langevin and Pascal Véron. Springer Verlag, vol. 3 (4), pp.113, 2011, Cryptography and Communications, ⟨10.1007/s12095-011-0056-0⟩. ⟨hal-00682554⟩
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Yves Aubry, Claude Carlet, Philippe Langevin, Pascal Véron. Cryptography and Communications. Springer, pp.113, 2011, ⟨10.1007/s12095-011-0056-0⟩. ⟨hal-00979704⟩
resume
This is a special issue of "Cryptography and Communications - Discrete Structures, Boolean functions and sequences" in the honor of Jacques Wolfmann.
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Pré-publication, Document de travail

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Yves Aubry, Jean-Christophe Godin, Olivier Togni. Choosability of a weighted path and free-choosability of a cycle. 2011. ⟨hal-00484445v3⟩
resume
A graph $G$ with a list of colors $L(v)$ and weight $w(v)$ for each vertex $v$ is $(L,w)$-colorable if one can choose a subset of $w(v)$ colors from $L(v)$ for each vertex $v$, such that adjacent vertices receive disjoint color sets. In this paper, we give necessary and sufficient conditions for a weighted path to be $(L,w)$-colorable for some list assignments $L$. Furthermore, we solve the problem of the free-choosability of a cycle.
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https://hal.science/hal-00484445/file/yajcgotJCMCC.pdf BibTex

2010

Article dans une revue

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Yves Aubry, François Rodier. Differentially 4-uniform functions. Contemporary mathematics, 2010, 521, pp.1--8. ⟨10.1090/conm/521⟩. ⟨hal-00403224⟩
resume
We give a geometric characterization of vectorial boolean functions with differential uniformity less or equal to 4.
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https://hal.science/hal-00403224/file/Diff.Unif.Func.9juillet.pdf BibTex
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Yves Aubry, Gary Mcguire, François Rodier. A few more functions that are not APN infinitely often. Contemporary mathematics, 2010, 518, pp.23--31. ⟨10.1090/conm/518⟩. ⟨hal-00415755v2⟩
resume
We consider exceptional APN functions on ${\bf F}_{2^m}$, which by definition are functions that are not APN on infinitely many extensions of ${\bf F}_{2^m}$. Our main result is that polynomial functions of odd degree are not exceptional, provided the degree is not a Gold member ($2^k+1$) or a Kasami-Welch number ($4^k-2^k+1$). We also have partial results on functions of even degree, and functions that have degree $2^k+1$.
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https://hal.science/hal-00415755/file/YAFRGMG.rerevised.pdf BibTex

Pré-publication, Document de travail

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Yves Aubry, Godin Jean-Christophe, Togni Olivier. Extended core and choosability of a graph. 2010. ⟨hal-00492157⟩
resume
A graph $G$ is $(a,b)$-choosable if for any color list of size $a$ associated with each vertices, one can choose a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. This paper shows an equivalence between the $(a,b)$-choosability of a graph and the $(a,b)$-choosability of one of its subgraphs called the extended core. As an application, this result allows to prove the $(5,2)$-choosability and $(7,3)$-colorability of triangle-free induced subgraphs of the triangular lattice.
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https://hal.science/hal-00492157/file/yves.jc.olivier.soumis.pdf BibTex

2008

Article dans une revue

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Yves Aubry, Régis Blache. On some questions related to the Gauss conjecture for function fields. Journal of Number Theory, 2008, 128 (7), pp.2053--2062. ⟨10.1016/j.jnt.2007.10.014⟩. ⟨hal-00978911⟩
resume
We show that, for any finite field Fq , there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.
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https://hal.science/hal-00978911/file/AubryBlache5_10.pdf BibTex
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Yves Aubry, Philippe Langevin. On the semiprimitivity of cyclic codes. On Number Theory and its Applications, 2008, 5, pp.284--293. ⟨10.1142/6761⟩. ⟨hal-00978910⟩
resume
We prove, without assuming the Generalized Riemann Hypothesis, but with at most one exception, that an irreducible cyclic code c(p,m,v) with v prime and p of index 2 modulo v is a two-weight code if and only if it is a semiprimitive code or it is one of the six sporadic known codes. The result is proved without any exception for index-two irreducible cyclic c(p,m,v) codes with v prime not congruent to 3 modulo 8. Finally, we prove that these two results hold true in fact for irreducible cyclic code c(p,m,v) such that there is three p-cyclotomic cosets modulo v.
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https://hal.science/hal-00978910/file/SAGA_2008.pdf BibTex

Communication dans un congrès

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Yves Aubry, Philippe Langevin. On the semiprimitivity of irreducible cyclic codes. Symposium on Algebraic Geometry and its Applications, 2008, Tahiti, Unknown Region. pp.284--293. ⟨hal-01279296⟩
resume
no abstract
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2006

Article dans une revue

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Yves Aubry. Class number in non Galois quartic and non abelian Galois octic function fields over finite fields. Bulletin of the Greek Mathematical Society, 2006, 52, pp.61--76. ⟨hal-00978906⟩
resume
We consider a totally imaginary extension of a real extension of a rational function field over a finite field of odd characteristic. We prove that the relative ideal class number one problem for such non Galois quartic fields is equivalent to the one for non abelian Galois octic imaginary functions fields. Then, we develop some results on characters which give a method to evaluate the ideal class number of such quartic function fields.
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https://hal.science/hal-00978906/file/BullGreekMathSoc_2006.pdf BibTex
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Yves Aubry, Philippe Langevin. On the weights of binary irreducible cyclic codes. Lecture Notes in Computer Science, 2006, 3969, pp.46--54. ⟨10.1007/11779360_5⟩. ⟨hal-00978908⟩
resume
This paper is devoted to the study of the weights of binary irreducible cyclic codes. We start from McEliece's interpretation of these weights by means of Gauss sums. Firstly, a dyadic analysis, using the Stickelberger congruences and the Gross-Koblitz formula, enables us to improve McEliece's divisibility theorem by giving results on the multiplicity of the weights. Secondly, in connection with a Schmidt and White's conjecture, we focus on binary irreducible cyclic codes of index two. We show, assuming the generalized Riemann hypothesis, that there are an infinite of such codes. Furthermore, we consider a subclass of this family of codes satisfying the quadratic residue conditions. The parameters of these codes are related to the class number of some imaginary quadratic number fields. We prove the non existence of such codes which provide us a very elementary proof, without assuming G.R.H, that any two-weight binary irreducible cyclic code c(m,v) of index two with v prime greater that three is semiprimitive.
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Communication dans un congrès

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Yves Aubry, Philippe Langevin. On the weight of binary irreducible cyclic codes. Workshop on Coding and Cryptography WCC'05, 2006, Unknown, Norway. pp.46--54. ⟨hal-01279323⟩
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2005

Communication dans un congrès

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Yves Aubry, Philippe Langevin. On the weight of binary irreducible cyclic codes. Book of Abstracts of the Workshop on Coding and Cryptography WCC'05, 2005, Bergen, Norway. pp.161--169. ⟨hal-01279324⟩
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no abstract
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Ouvrages

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Yves Aubry, Gilles Lachaud. Arithmetic, Geometry and Coding Theory. Société Mathématique de France. Société Mathématique de France, pp.216, 2005, Séminaires et congrès, 2-85629-175-9. ⟨hal-00979006⟩
resume
In may 2003, two events have been held in the ''Centre International de Rencontres Mathématiques'' in Marseille (France), devoted to Arithmetic, Geometry and their applications in Coding theory and Cryptography: an European school ''Algebraic Geometry and Information Theory'' and the 9-th international conference ''Arithmetic, Geometry and Coding Theory''. Some of the courses and the conferences are published in this volume. The topics were theoretical for some ones and turned towards applications for others: abelian varieties, function fields and curves over finite fields, Galois group of pro-p-extensions, Dedekind zeta functions of number fields, numerical semigroups, Waring numbers, bilinear complexity of the multiplication in finite fields and class number problems.
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2004

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Yves Aubry, Marc Perret. On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields. Finite Fields and Their Applications, 2004, 10 (3), pp.412--431. ⟨10.1016/j.ffa.2003.09.005⟩. ⟨hal-00978903⟩
resume
We give a formula for the number of rational points of projective algebraic curves defined over a finite field, and a bound ''á la Weil'' for connected ones. More precisely, we give the characteristic polynomials of the Frobenius endomorphism on the étale l-adic cohomology groups of the curve. Finally, as an analogue of Artin's holomorphy conjecture, we prove that, if $Y\longrightarrow X$ is a finite flat morphism between two varieties over a finite field, then the characteristic polynomial of the Frobenius morphism on the i-th l-adic cohomology group with compact support of $X$ divides that of $Y$ for any i. We are then enable to give an estimate for the number of rational points in a flat covering of curves.
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https://hal.science/hal-00978903/file/FFTA_2004_Version_Auteurs.pdf BibTex
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Yves Aubry, Marc Perret. Divisibility of zeta functions of curves in a covering. Archiv der Mathematik, 2004, 82, pp.205--213. ⟨hal-00978051⟩
resume
As an analogous of a conjecture of Artin, we show that, if $ Y\longrightarrow X$ is a finite flat morphism between two singular reduced absolutely irreducible projective algebraic curves defined over a finite field, then the numerator polynomial of the zeta function of $X$ divides those of $Y$ in ${\sb Z}[T]$. We give some interpretations of this result in terms of semi-abelian varieties.
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https://hal.science/hal-00978051/file/ArchivDerMath2004.pdf BibTex

2002

HDR

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Yves Aubry. Variétés algébriques et corps de fonctions sur un corps fini. Géométrie algébrique [math.AG]. Aix-Marseille Université, 2002. ⟨tel-00977396⟩
resume
Nous nous intéressons au nombre de points rationnels des variétés algébriques projectives sur un corps fini. Nous déterminons notamment la fonction zêta (et plus précisément les polynômes caractéristiques de l'endomorphisme de Frobenius sur les espaces de cohomologie étale l-adique) des courbes algébriques projectives sans autre hypothèse de lissité ou d'irréductibilité. Nous montrons la divisibilité de ces polynômes dans un revêtement plat de courbes connexes, que l'on peut interpréter comme un analogue de la conjecture d'holomorphie d'Artin sur les fonctions zêta de Dedekind des corps de nombres. Nous obtenons des bornes sur le nombre de points rationnels sur un corps fini dans un revêtement plat entre courbes algébriques projectives connexes, généralisant les bornes connues et notamment celle de Weil. Nous nous sommes également intéressé au problème du nombre de classes dans les corps de fonctions à une variable sur un corps fini. Nous avons établi un théorème de finitude en ce qui concerne les extensions totalement imaginaires d'extensions totalement réelles dont le nombre de classes d'idéaux du corps imaginaire est fixé . Dans le cas où ces extensions sont quadratiques, nous donnons une formule du nombre de classes relatif en terme de fonction L, ainsi qu'une formule liant cette fonction L à une somme de caractères de type Legendre dans le cas du nombre de classe 1. Si l'on suppose de plus que le groupe de Galois d'une telle extension est isomorphe au groupe de Klein, via la théorie du corps de classes ainsi que des factorisations de fonctions zêta et des estimations de régulateurs, nous déterminons ces corps via les extensions d'Artin-Schreier et les jacobiennes.
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HDR
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https://theses.hal.science/tel-00977396/file/HDR-Finale.pdf BibTex

1999

Article dans une revue

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Yves Aubry, Dominique Le Brigand. Imaginary bicyclic biquadratic function fields in characteristic two. Journal of Number Theory, 1999, 77, pp.36--50. ⟨hal-00977317⟩
resume
We are interested in the analogue of a result proved in the number field case by E. Brown and C.J. Parry and in the function field case in odd characteristic by Zhang Xianke. Precisely, we study the ideal class number one problem for imaginary quartic Galois extensions of $k=\fq(x)$ of Galois group $\integers /2\integers \times\integers /2\integers$ in even characteristic.\\Let $L/k$ be such an extension and let $K_1$, $K_2$ and $K_3$ be the distinct subfields extensions of $L/k$. In even characteristic, the fields $K_i$ are Artin-Schreier extensions of $k$ and $L$ is the compositum of any two of them.\\Using the factorization of the zeta functions of this fields, we get a formula between their ideal class numbers which enables us to find all imaginary quartic Galois extensions $L/k$ of Galois group $\integers /2\integers \times\integers /2\integers$ with ideal class number one.
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https://hal.science/hal-00977317/file/J.NumberTheory1999.pdf BibTex

1996

Article dans une revue

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Yves Aubry. Class number in totally imaginary extensions of totally real function fields. Lecture Notes Series London Mathematical Society, 1996, Proceedings of the Third International Conference on Finite fields and Applications, pp.23--29. ⟨hal-00977316⟩
resume
We show that, up to isomorphism, there are only finitely many totally real function fields which have any totally imaginary extension of a given ideal class number.
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https://hal.science/hal-00977316/file/ImaginaryExtension_Glasgow_.pdf BibTex
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Yves Aubry, Marc Perret. A Weil theorem for singular curves. Contemporary mathematics, 1996, Walter de Gruyter, pp.1--8. ⟨10.1515/9783110811056.1⟩. ⟨hal-00976485⟩
resume
We generalize Weil's theorem on the number of rational points of smooth curves over a finite field to singular ones.
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https://hal.science/hal-00976485/file/Weil.Singulier.pdf BibTex

1995

Article dans une revue

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Yves Aubry, Marc Perret. Coverings of singular curves over finite fields. Manuscripta mathematica, 1995, 88 (1), pp.467--478. ⟨10.1007/BF02567835⟩. ⟨hal-00976489⟩
resume
We prove that if $f : Y\longrightarrow X$ is a finite fiat morphism between two reduced absolutely irreducible algebraic projective curves defined over the finite field ${\sb F}_q$, then $$\mid \sharp Y({\sb F}_q) - \sharp X({\sb F}_q)\mid \leq 2({\pi}_Y - {\pi}_X)\sqrt q,$$ where $\pi_C$ is the arithmetic genus of a curve $C$. As application, we give some character sum estimation on singular curves.
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https://hal.science/hal-00976489/file/ManuscriptaMathematica1995.pdf BibTex

1992

Article dans une revue

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Yves Aubry. Reed-Muller codes associated to projective algebraic varieties. Lecture Notes in Mathematics, 1992, Algebraic Geometry and Coding Theory, 1518, pp.4-17. ⟨10.1007/BFb0087988⟩. ⟨hal-00973776⟩
resume
The classical generalized Reed-Muller codes introduced by Kasami, Lin and Peterson [5], and studied also by Delsarte, Goethals and Mac Williams [2], are defined over the affine space An(Fq) over the finite field Fq with q elements. Moreover Lachaud [6], following Manin and Vladut [7], has considered projective Reed-Muller codes, i.e. defined over the projective space Pn(Fq). In this paper, the evaluation of the forms with coefficients in the finite field Fq is made on the points of a projective algebraic variety V over the projective space Pn(Fq). Firstly, we consider the case where V is a quadric hypersurface, singular or not, Parabolic, Hyperbolic or Elliptic. Some results about the number of points in a (possibly degenerate) quadric and in the hyperplane sections are given, and also is given an upper bound of the number of points in the intersection of two quadrics. In application of these results, we obtain Reed-Muller codes of order 1 associated to quadrics with three weights and we give their parameters, as well as Reed-Muller codes of order 2 with their parameters. Secondly, we take V as a hypersurface, which is the union of hyperplanes containing a linear variety of codimension 2 (these hypersurfaces reach the Serre bound). If V is of degree h, we give parameters of Reed-Muller codes of order d < h, associated to V.
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https://inria.hal.science/hal-00973776/file/Aubry_LectureNotesInMaths.pdf BibTex

Pré-publication, Document de travail

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Yves Aubry. Algebraic geometric codes on surfaces. 1992. ⟨hal-00979000⟩
resume
For a given algebraic variety $V$ defined over a finite field and a very ample divisor $D$ on $V$, we give a construction of a linear code $C_{V,D}$. If $V$ is a curve, we recover the algebraic geometric Goppa codes. We are interested here in the case where $V$ is an algebraic surface, and we give in some cases the parameters of such corresponding codes. We compare these parameters to the Singleton bound and to those of Goppa codes. In order to compute these parameters, we use the Riemann-Roch theorem for surfaces.
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Pré-publication, Document de travail
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https://hal.science/hal-00979000/file/EuroCode.1992.pdf BibTex