29 research outputs found
Generalized Hamming weights of affine cartesian codes
In this article, we give the answer to the following question: Given a field
, finite subsets of , and linearly
independent polynomials of total
degree at most . What is the maximal number of common zeros
can have in ? For , the
finite field with elements, answering this question is equivalent to
determining the generalized Hamming weights of the so-called affine Cartesian
codes. Seen in this light, our work is a generalization of the work of
Heijnen--Pellikaan for Reed--Muller codes to the significantly larger class of
affine Cartesian codes.Comment: 12 Page
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 and a non-degenerate Hermitian
surface in \PP^3(\Fqt). The conjecture was proven to be true by Edoukou in
the case when . In this paper, we prove that the conjecture is true for
and . 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 -rational points on nonsingular threefolds in
We determine the maximum number of -rational points that a
nonsingular threefold of degree in a projective space of dimension
defined over 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
Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields
About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the
maximum number of common zeros that linearly independent homogeneous
polynomials of degree in variables with coefficients in a finite
field with elements can have in the corresponding -dimensional
projective space. Recently, it has been shown by Datta and Ghorpade that this
conjecture is valid if is at most and can be invalid otherwise.
Moreover a new conjecture was proposed for many values of beyond . In
this paper, we prove that this new conjecture holds true for several values of
. In particular, this settles the new conjecture completely when . Our
result also includes the positive result of Datta and Ghorpade as a special
case. Further, we determine the maximum number of zeros in certain cases not
covered by the earlier conjectures and results, namely, the case of and
of . All these results are directly applicable to the determination of the
maximum number of points on sections of Veronese varieties by linear
subvarieties of a fixed dimension, and also the determination of generalized
Hamming weights of projective Reed-Muller codes.Comment: 15 page
Two-Point Codes for the Generalized GK curve
We improve previously known lower bounds for the minimum distance of certain
two-point AG codes constructed using a Generalized Giulietti-Korchmaros curve
(GGK). Castellanos and Tizziotti recently described such bounds for two-point
codes coming from the Giulietti-Korchmaros curve (GK). Our results completely
cover and in many cases improve on their results, using different techniques,
while also supporting any GGK curve. Our method builds on the order bound for
AG codes: to enable this, we study certain Weierstrass semigroups. This allows
an efficient algorithm for computing our improved bounds. We find several new
improvements upon the MinT minimum distance tables.Comment: 13 page
Vanishing ideals of projective spaces over finite fields and a projective footprint bound
We consider the vanishing ideal of a projective space over a finite field. An
explicit set of generators for this ideal has been given by Mercier and
Rolland. We show that these generators form a universal Gr\"obner basis of the
ideal. Further we give a projective analogue of the footprint bound, and a
version of it that is suitable for estimating the number of points of a
projective algebraic variety over a finite field. An application to Serre's
inequality for the number of rational points of projective hypersurfaces over
finite fields is includedComment: 16 pages, slightly revised version, to appear in Acta Math. Sin.
(Engl. Ser.