Special numbers of rational points on hypersurfaces in the nn-dimensional projective space over a finite field

We study first some arrangements of hyperplanes in the nn-dimensional projective space Pn(Fq)Pn(Fq). Then we compute, in particular, the second and the third highest numbers of rational points on hypersurfaces of degree dd. As an application of our results, we obtain some weights of the Generalized Projective Reed–Muller codes PRM(q,d,n)PRM(q,d,n). We also list all the homogeneous polynomials reaching such numbers of zeros and giving the correspondent weights.