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

Abstract

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.