Comptes Rendus
Théorie des nombres
Cyclotomie des sommes de Weil binomiales
[Cyclotomy of Weil sums of binomials]
Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 373-376.

Weil sums of the form WK,d(a)=xKψ(xd+ax), where K is a finite field, ψ is an additive character of K, d is coprime to |K×|, and aK×, arise often in number theory, as well as in finite geometry, in cryptography, in the study of the correlation of sequences, and in coding theory. Here we are interested in the case where WK,d(a) takes only three distinct values as a runs through K×. Via a Galois-theoretic approach, we give several results concerning three-valued Weil sums, and, in particular, we generalize to any nonzero characteristic some results of Calderbank–McGuire–Poonen–Rubinstein, of Calderbank–McGuire and of Charpin proved in characteristic 2.

Les sommes de Weil de la forme WK,d(a)=xKψ(xd+ax), où K est un corps fini, ψ un caractère additif de K, d un entier premier à |K×| et aK×, apparaissent naturellement en théorie des nombres ainsi qu'en géométrie finie, en cryptographie, dans l'étude de la corrélation des suites et en théorie des codes. Nous nous intéressons ici au cas où WK,d(a) ne prend que trois valeurs distinctes lorsque a varie dans K×. Via une approche galoisienne, nous donnons plusieurs résultats concernant ces sommes de Weil à trois valeurs, généralisant notamment à toute caractéristique non nulle des résultats de Calderbank–McGuire–Poonen–Rubinstein, de Calderbank–McGuire et de Charpin établis en caractéristique 2.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2014.03.001
Yves Aubry 1, 2; Daniel J. Katz 3; Philippe Langevin 1

1 Institut de mathématiques de Toulon, Université de Toulon, France
2 Institut de mathématiques de Marseille, Aix-Marseille Université – CNRS, France
3 Department of Mathematics, California State University, Northridge, United States
@article{CRMATH_2014__352_5_373_0,
     author = {Yves Aubry and Daniel J. Katz and Philippe Langevin},
     title = {Cyclotomie des sommes de {Weil} binomiales},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {373--376},
     publisher = {Elsevier},
     volume = {352},
     number = {5},
     year = {2014},
     doi = {10.1016/j.crma.2014.03.001},
     language = {fr},
}
TY  - JOUR
AU  - Yves Aubry
AU  - Daniel J. Katz
AU  - Philippe Langevin
TI  - Cyclotomie des sommes de Weil binomiales
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 373
EP  - 376
VL  - 352
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2014.03.001
LA  - fr
ID  - CRMATH_2014__352_5_373_0
ER  - 
%0 Journal Article
%A Yves Aubry
%A Daniel J. Katz
%A Philippe Langevin
%T Cyclotomie des sommes de Weil binomiales
%J Comptes Rendus. Mathématique
%D 2014
%P 373-376
%V 352
%N 5
%I Elsevier
%R 10.1016/j.crma.2014.03.001
%G fr
%F CRMATH_2014__352_5_373_0
Yves Aubry; Daniel J. Katz; Philippe Langevin. Cyclotomie des sommes de Weil binomiales. Comptes Rendus. Mathématique, Volume 352 (2014) no. 5, pp. 373-376. doi : 10.1016/j.crma.2014.03.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.03.001/

[1] Y. Aubry; D.J. Katz; P. Langevin Cyclotomy of Weil sums of binomials, 2014 | arXiv

[2] A.R. Calderbank; G. McGuire Proof of a conjecture of Sarwate and Pursley regarding pairs of binary m-sequences, IEEE Trans. Inf. Theory, Volume 41 (1995) no. 4, pp. 1153-1155

[3] A.R. Calderbank; G. McGuire; B. Poonen; M. Rubinstein On a conjecture of Helleseth regarding pairs of binary m-sequences, IEEE Trans. Inf. Theory, Volume 42 (1996) no. 3, pp. 988-990

[4] P. Charpin Cyclic codes with few weights and Niho exponents, J. Comb. Theory, Ser. A, Volume 108 (2004) no. 2, pp. 247-259

[5] H. Davenport; H. Hasse Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. Reine Angew. Math., Volume 172 (1935), pp. 151-182

[6] T. Feng On cyclic codes of length 22r with two zeros whose dual codes have three weights, Des. Codes Cryptogr., Volume 62 (2012) no. 3, pp. 253-258

[7] T. Helleseth Some results about the cross-correlation function between two maximal linear sequences, Discrete Math., Volume 16 (1976) no. 3, pp. 209-232

[8] D.J. Katz Weil sums of binomials, three-level cross-correlation, and a conjecture of Helleseth, J. Comb. Theory, Ser. A, Volume 119 (2012) no. 8, pp. 1644-1659

[9] N. Katz; R. Livné Sommes de Kloosterman et courbes elliptiques universelles en caractéristiques 2 et 3, C. R. Acad. Sci. Paris, Ser. I, Volume 309 (1989) no. 11, pp. 723-726

[10] G. Lachaud; J. Wolfmann Sommes de Kloosterman, courbes elliptiques et codes cycliques en caractéristique 2, C. R. Acad. Sci. Paris, Ser. I, Volume 305 (1987) no. 20, pp. 881-883

[11] Y. Niho Multi-valued cross-correlation function between two maximal linear recursive sequences, University of Southern California, Los Angeles, USA, 1972 (PhD thesis)

[12] D.V. Sarwate; M.B. Pursley Cross correlation properties of pseudorandom and related sequences, IEEE Trans. Inf. Theory, Volume 68 (1980) no. 5, pp. 593-619 (Correction dans IEEE Trans. Inf. Theory, 68, 12, 1980, pp. 1554)

Cited by Sources:

Comments - Policy