de Mathématiques de Luminy
Special numbers of rational points on hypersurfaces
in the n-dimensional projective space over a finite field.
We study first some arrangements of hyperplanes in the n-dimensional projective space . Then we compute, in particular, the second and the third highest numbers of rational points on hypersurfaces of degree d. As application of our results we obtain some weights of the Generalized Pro jective Reed-Muller codes PRM(q,d,n). And we also list all the homogeneous polynomials reaching such numbers of zeros and giving the correspondent weights.
2000 Mathematics Subject Classification. 11T71, 14J70, 05B25.