International audienceWe generalize Weil's theorem on the number of rational points of smooth curves over a finite field to singular ones
We consider exceptional APN functions on F 2 m , which by definition are functions that are APN on infinitely many extensions of F 2 m . Our main result is that polynomial functions of odd degree are not exceptional, provided the degree is not a Gold number (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.
International audienceThe 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
Abstract. 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.
Abstract. The Weil sum W K,d (a) = x∈K ψ(x d + ax) where K is a finite field, ψ is an additive character of K, d is coprime to |K × |, and a ∈ K × 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 × . A Galois-theoretic approach, combined with p-divisibility results on Gauss sums, 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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