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 F,, 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 F-q are given only by some hyperplane arrangements. We show that for q/2 + 5/2 <= d < q, this is no longer the case: the third highest number associated to some hyperplane arrangements can also be obtained in this case by some hypersurface, containing an irreducible quadric. For the curves on FP with p a prime number we show that this condition is the best possible. (c) 2008 Elsevier Inc. All rights reserved.