On maximal and minimal hypersurfaces of Fermat type (original) (raw)
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Moscow Mathematical Journal, 2015
We give a short proof of an inequality, conjectured by Tsfasman and proved by Serre, for the maximum number of points of hypersurfaces over finite fields. Further, we consider a conjectural extension, due to Tsfasman and Boguslavsky, of this inequality to an explicit formula for the maximum number of common solutions of a system of linearly independent multivariate homogeneous polynomials of the same degree with coefficients in a finite field. This conjecture is shown to be false, in general, but is also shown to hold in the affirmative in a special case. Applications to generalized hamming weights of projective Reed-Muller codes are outlined and a comparison with an older conjecture of Lachaud and a recent result of Couvreur is given.
arXiv (Cornell University), 2015
We give a short proof of an inequality, conjectured by Tsfasman and proved by Serre, for the maximum number of points of hypersurfaces over finite fields. Further, we consider a conjectural extension, due to Tsfasman and Boguslavsky, of this inequality to an explicit formula for the maximum number of common solutions of a system of linearly independent multivariate homogeneous polynomials of the same degree with coefficients in a finite field. This conjecture is shown to be false, in general, but is also shown to hold in the affirmative in a special case. Applications to generalized hamming weights of projective Reed-Muller codes are outlined and a comparison with an older conjecture of Lachaud and a recent result of Couvreur is given.
Discrete Mathematics, 2009
We study first some arrangements of hyperplanes in the n-dimensional projective space P n (F q). 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 Projective Reed-Muller codes P RM (q, d, n). And we also list all the homogeneous polynomials reaching such numbers of zeros and giving the correspondent weights.
Second highest number of points of hypersurfaces in Fqn
Finite Fields and Their Applications, 2007
For 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 F n q , 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.
On the Number of Points of Some Hypersurfaces in F n q
Finite Fields and Their Applications - FINITE FIELDS THEIR APPL, 1996
For generalized Reed–Muller codes, whenqis large enough, we give the second codeword weight, that is, the weight which is just above the minimal distance, and we also list all the codewords which reach this weight. To do this we have to study the number of points of some hypersurfaces and some arrangements of hyperplanes. We also present some properties of the Möbius function of these arrangements.
On algebraic curves over a finite field with many rational points
Bulletin of the Belgian Mathematical Society - Simon Stevin, 2000
In [12], a new upper bound for the number of F q-rational points on an absolutely irreducible algebraic plane curve defined over a finite field F q of degree d < √ q − 2 was obtained. The present paper is a continuation of [12] and the main result is a similar upper bound for the case d = √ q − 2.
On the number of points of algebraic sets over finite fields
Journal of Pure and Applied Algebra, 2015
We determine upper bounds on the number of rational points of an affine or projective algebraic set defined over an extension of a finite field by a system of polynomial equations, including the case where the algebraic set is not defined over the finite field by itself. A special attention is given to irreducible but not absolutely irreducible algebraic sets, which satisfy better bounds. We study the case of complete intersections, for which we give a decomposition, coarser than the decomposition in irreducible components, but more directly related to the polynomials defining the algebraic set. We describe families of algebraic sets having the maximum number of rational points in the affine case, and a large number of points in the projective case. Nous déterminons des majorations du nombre de points d'un ensemble algébrique affine ou projectif, défini sur une extension d'un corps fini par un système d'équations polynomiales, y compris dans le cas où l'ensemble algébrique n'est pas défini sur le corps fini lui-même. Une attention particulière est portée aux ensemble algébriques irréductibles mais non absolument irréductibles, pour lesquels nous obtenons de meilleures bornes. Nous étudions le cas des intersections complètes, pour lesquelles nous construisons une décomposition moins fine que la décomposition en composantes irréductibles, mais plus directement liée aux polynômes qui définissent l'ensemble algébrique. Enfin, nous construisons des familles d'ensembles algébriques atteignant le nombre maximum de points rationnels dans le cas affine, et comportant de nombreux points dans le cas projectifs.
Rational Points on Certain Hyperelliptic Curves over Finite Fields
Bulletin of the Polish Academy of Sciences Mathematics, 2007
Let K be a field, a, b ∈ K and ab = 0. Let us consider the polynomials g1(x) = x n + ax + b, g2(x) = x n + ax 2 + bx, where n is a fixed positive integer. In this paper we show that for each k ≥ 2 the hypersurface given by the equation
Second highest number of points of hypersurfaces in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll">mml:msubsup<mml:mi mathvariant="double-struck">Fmml:miqmml:min
Finite Fields and Their Applications, 2007
For 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 F n q , 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.