Abstract
The weight distribution of the generalized Reed–Muller codes over the finite field F q 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 F p with p a prime number we show that this condition is the best possible.
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