Second highest number of points of hypersurfaces in Fqn

AbstractFor generalized Reed–Muller, GRM(q,d,n), codes, the determination of the second weight is still generally unsolved, it is an open question of Cherdieu and Rolland [J.P. Cherdieu, R. Rolland, On the number of points of some hypersurfaces in Fqn, Finite Fields Appl. 2 (1996) 214–224]. In order to answer this question, we study some maximal hypersurfaces and we compute the second weight of GRM(q,d,n) codes with the restriction that q⩾2d

Remarks on low weight codewords of generalized affine and projective Reed-Muller codes

We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes. In the affine case we prove that if the size of the working finite field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then in the general case we study some types of codewords and prove that they cannot be second, thirds or fourth weight depending on the hypothesis. In the projective case the second distance of generalized Reed-Muller codes is estimated, namely a lower bound and an upper bound of this weight are given.Comment: New version taking into account recent results from Elodie Leducq on the characterization of the next-to-minimal codewords (cf. arXiv:1203.5244

Maximum number of points on intersection of a cubic surface and a non-degenerate Hermitian surface

In 1991 S{\o}rensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree $d$ and a non-degenerate Hermitian surface in \PP^3(\Fqt). The conjecture was proven to be true by Edoukou in the case when $d=2$. In this paper, we prove that the conjecture is true for $d=3$ and $q \ge 8$. We further determine the second highest number of rational points on the intersection of a cubic surface and a non-degenerate Hermitian surface. Finally, we classify all the cubic surfaces that admit the highest and second highest number of points in common with a non-degenerate Hermitian surface. This classifications disproves one of the conjectures proposed by Edoukou, Ling and Xing

Maximum number of Fq\mathbb{F}_q-rational points on nonsingular threefolds in P4\mathbb{P}^4

We determine the maximum number of $\mathbb{F}_q$-rational points that a nonsingular threefold of degree $d$ in a projective space of dimension $4$ defined over $\mathbb{F}_q$ may contain. This settles a conjecture by Homma and Kim concerning the maximum number of points on a hypersurface in a projective space of even dimension in this particular case.Comment: 8 page