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It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in P 4 (F q 2 ) has at most d (q 5 + q 2) + q 3 + 1 points in common with a threefold of degree d defined over F q 2 . He proved the conjecture for d = 2. In this paper, we show that the conjecture is true for d = 3 and q ≥ 7. [ABSTRACT FROM AUTHOR]

We determine the maximum number of F q -rational points that a nonsingular threefold of degree d in a projective space of dimension 4 defined over 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. [ABSTRACT FROM AUTHOR]

Let F be any field and A 1 , … , A m be finite subsets of F . We determine the maximum number of common zeroes a linearly independent family of r polynomials of degree at most d of F [ x 1 , … , x m ] can have in A 1 × … × A m . In the case when F is a finite field, our results resolve the problem of determining the generalized Hamming weights of affine Cartesian codes. This is a generalization of the work of Heijnen and Pellikaan where these were determined for the generalized Reed–Muller codes. Finally, we determine the duals of affine Cartesian codes and compute their generalized Hamming weights as well. [ABSTRACT FROM AUTHOR]