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⟩
- typdoc
- 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.
- typdoc
- Article dans une revue
- DOI
- DOI : 10.1007/s12095-023-00652-4
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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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-
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.
- typdoc
- Pré-publication, Document de travail
- Accès au texte intégral et 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.
- typdoc
- Article dans une revue
- DOI
- DOI : 10.1017/S0017089510000017
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2005
Communication dans un congrès
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- 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.
- typdoc
- Communication dans un congrès
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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.
- typdoc
- 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
- typdoc
- 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].
- typdoc
- Chapitre d'ouvrage
- DOI
- DOI : 10.1007/978-3-7091-2786-5_12
- Accès au bibtex
-