Bounds on the number of rational points of algebraic hypersurfaces over finite fields, with applications to projective Reed-Muller codes
Daniele Bartoli, Adnen Sboui, Leo Storme · Advances in Mathematics of Communications · 2016
We present bounds on the number of points in algebraic curves and algebraic hypersurfaces in $\mathbb{P}^n(\mathbb{F}_q)$ of small degree $d$, depending on the number of linear components contained in such curves and hypersurfaces. The obtained results have applications to the weight distribution of the projective Reed-Muller codes PRM$(q,d,n)$ over the finite field $\mathbb{F}_q$.