Highest numbers of points of hypersurfaces over finite fields and generalized Reed–Muller codes

The weight distribution of the generalized Reed–Muller codes over the finite field Fq is linked to the number of points of some hypersurfaces of degree d in the n-dimensional space over the same field. For d⩽q/3+2, the three first highest numbers of points of hypersurfaces of degree d in the n-dimensional projective space over the finite field Fq are given only by some hyperplane arrangements. We show that for q/2+5/2⩽d