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

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