Gilles Lachaud

Higher Weights of Grassmann Codes

Using a combinatorial approach to studying the hyperplane sections of Grassmannians, we give two new proofs of a result of Nogin concerning the higher weights of Grassmann codes. As a consequence, we obtain a bound on the number of higher dimensional subcodes of the Grassmann code having the minimum Hamming norm. We also discuss a generalization of Grassmann codes.

Number of solutions of equations over finite fields and a conjecture of Lang and Weil

A brief survey of the conjectures of Weil and some classical estimates for the number of points of varieties over finite fields is given. The case of partial flag manifolds is discussed in some details by way of an example. This is followed by a motivated account of some recent results on counting the number of points of varieties over finite fields, and a related conjecture of Lang and Weil. Explicit combinatorial formulae for the Betti numbers and the Euler characteristics of smooth complete intersections are also discussed.

Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory

We consider the question of determining the maximum number of \(\mathbb{F}_{q}\)-rational points that can lie on a hypersurface of a given degree in a weighted projective space over the finite field \(\mathbb{F}_{q}\), 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 \(\mathbb{F}_{q}\). 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.

On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius

The purpose of this article is to study the distribution of the trace on the unitary symplectic group. We recall its relevance to equidistribution results for the eigenvalues of the Frobenius in families of abelian varieties over finite fields, and to the limiting distribution of the number of points of curves. We give four expressions of the trace distribution if g = 2, in terms of special functions, and also an expression of the distribution of the trace in terms of elementary symmetric functions. In an appendix, we prove a formula for the trace of the exterior power of the identity representation.