Publications HAL de valerie,Gillot

2024

Article dans une revue

titre
Covering radius of $ RM(4,8) $
auteur
Valérie Gillot, Philippe Langevin
article
Advances in Mathematics of Communications, 2024, 18 (2), pp.383-393. ⟨10.3934/amc.2023038⟩
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Article dans une revue
DOI
DOI : 10.3934/amc.2023038
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2023

Article dans une revue

titre
Classification of some cosets of the Reed-Muller code
auteur
Valérie Gillot, Philippe Langevin
article
Cryptography and Communications - Discrete Structures, Boolean Functions and Sequences , 2023, 15 (6), pp.1129-1137. ⟨10.1007/s12095-023-00652-4⟩
resume
This paper presents a descending method to classify Boolean functions in 7 variables under the action of the affine general linear group. The classification determines the number of classes, a set of orbits representatives and a generator set of the stabilizer of each representative. The method consists in the iteration of the classification process of RM(k, m) / RM(r- 1 , m) from that of RM(k, m)/RM(r, m). We namely obtain the classifications of RM(4, 7)/RM(2, 7) and of RM(7, 7)/RM(3, 7), from which we deduce some consequences on the covering radius of RM(3, 7) and the classification of near bent functions.
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Article dans une revue
DOI
DOI : 10.1007/s12095-023-00652-4
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https://hal.science/hal-04209845/file/2208.02469.pdf BibTex

2022

Communication dans un congrès

titre
CLASSIFICATION OF BOOLEAN FUNCTIONS
auteur
Valérie Gillot, Philippe Langevin
article
Boolean Function and their Applications, Sep 2022, Balestrand, Norway
resume
This note presents a descending method that allows us to classify quotients of Reed-Muller codes of lenghth 128 under the action of the affine general linear group.
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Communication dans un congrès
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https://univ-tln.hal.science/hal-03740160/file/aglclass.pdf BibTex

Pré-publication, Document de travail

titre
CLASSIFICATION OF SOME COSETS OF THE REED-MULLER CODE
auteur
Valérie Gillot, Philippe Langevin
article
2022
resume
This paper presents a descending method to classify Boolean functions in 7 variables under the action of the affine general linear group. The classification determines the number of classes, a set of orbits representatives and a generator set of the stabilizer of each representative. The method consists in the iteration of the classification process of RM (k, m)/RM (r − 1, m) from that of RM (k, m)/RM (r, m). We namely obtain the classifications of RM (4, 7)/RM (2, 7) and of RM (7, 7)/RM (3, 7), from which we deduce some consequences on the covering radius of RM (3, 7) and the classification of near bent functions.
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Pré-publication, Document de travail
Accès au texte intégral et bibtex
https://univ-tln.hal.science/hal-03834481/file/BFA-paper2022.pdf BibTex

2010

Article dans une revue

titre
ESTIMATION OF SOME EXPONENTIAL SUM BY MEANS OF q-DEGREE
auteur
Valérie Gillot, Philippe Langevin
article
Glasgow Mathematical Journal, 2010, 52, pp.315--324. ⟨10.1017/S0017089510000017⟩
resume
In this paper, we improve results of Gillot, Kumar and Moreno to estimate some exponential sums by means of q-degrees. The method consists in applying suitable elementary transformations to see an exponential sum over a finite field as an exponential sum over a product of subfields in order to apply Deligne bound. In particular, we obtain new results on the spectral amplitude of some monomials.
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Article dans une revue
DOI
DOI : 10.1017/S0017089510000017
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https://hal.science/hal-00993148/file/asm.pdf BibTex

2005

Communication dans un congrès

titre
Exponential Sums and Boolean Functions
auteur
Julien Bringer, Valérie Gillot, Philippe Langevin
article
Proceedings of BFCA'05 Conference, March 7--8, 2005 Rouen, France, Mar 2005, Rouen, France. pp.177--186
resume
We study the nonlinearity of Boolean functions constructed by means of a subgroup of the multiplicative group of a finite field. The functions that we consider are constant over the non trivial cosets of a subgroup of small index. Classical properties of Gauss sums lead us to propose a new conjecture of the Patterson-Wiedemann type. One of the major steps of this approach consists in finding good estimations of exponential sums restricted over subgroup.
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Communication dans un congrès
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https://hal.science/hal-00993895/file/sebf.pdf BibTex

1995

Article dans une revue

titre
Bounds for Exponential Sums over Finite Fields
auteur
Valérie Gillot
article
Finite Fields and Their Applications, 1995, 1 (4), pp.421--436. ⟨10.1006/ffta.1995.1033⟩
resume
In this paper, bounds for exponential sums associated to polynomial @? defined over finite fields are given. We introduce the q-ary weight of the degrees of the polynomial @? and the bound is expressed in particular in terms of these when the usual expression depends on the degree. To do this, we associate to @? a transformed polynomial F"R, we apply the Deligne bound to this polynomial, and we extend the results established by O. Moreno and P. Kumar for other polynomial families @? and for fields of any characteristic. The bound obtained here improves the famous Weil bound in several cases. In the same way, using results of A. Adolphson and S. Sperber involving Newton polyhedra, bounds for diagonal polynomials are also given in terms of q-ary weight of their degrees, when the underlying field is a quadratic extension of F"q. Finally, we apply the previous result to obtain a bound of the number of zeros of diagonal polynomials.
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Article dans une revue
DOI
DOI : 10.1006/ffta.1995.1033
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1994

Communication dans un congrès

titre
Minimum Weight for Codes Stemmed from Exponential Sums Bounds
auteur
Valérie Gillot
article
the 15-th Symposium on Information Theory in the Benelux, 1994, Belgium. pp.53-60
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Communication dans un congrès
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1993

Chapitre d'ouvrage

titre
Bound for Trace-Equation and Application to Coding Theory
auteur
Valérie Gillot
article
Camion, P. and Charpin, P. and Harari, S. Eurocode '92, International Symposium on Coding Theory and Applications, 339, Springer Vienna, pp.135-146, 1993, 978-3-211-82519-8. ⟨10.1007/978-3-7091-2786-5_12⟩
resume
Using the vector space structure of finite field extension of characteristic two, we transform a one-variable polynomial into a two-variable one. Applying the Deligne bound about exponential sums [1] to the two-variable polynomial, we improve in several cases the bound on the number of solutions for trace-equation. In this way, we obtain different results in Coding Theory. Using the trace-description of the codewords for cyclic codes introduced by Wolfmann [6], we improve the bound on weights for these codes and we also generalize, in different cases, the results of Moreno and Kumar in [2].
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Chapitre d'ouvrage
DOI
DOI : 10.1007/978-3-7091-2786-5_12
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