13 research outputs found
Complete arcs arising from a generalization of the Hermitian curve
We investigate complete arcs of degree greater than two, in projective planes
over finite fields, arising from the set of rational points of a generalization
of the Hermitian curve. The degree of the arcs is closely related to the number
of rational points of a class of Artin-Schreier curves which is calculated by
using exponential sums via Coulter's approach. We also single out some examples
of maximal curves
An elementary abelian -cover of the Hermitian curve with many automorphisms
A certain elementary abelian -cover of the Hermitian curve with many
automorphisms in characteristic is considered. It is proved that the
order of Sylow -groups of the automorphism group of this curve is close to
Nakajima's bound in terms of the -rank. Weierstrass points, Galois points,
Frobenius nonclassicality, and the arc property are also investigated.Comment: 9 page
Complete (k,3)-arcs from quartic curves
Complete (Formula presented.) -arcs in projective planes over finite fields are the geometric counterpart of linear non-extendible Near MDS codes of length (Formula presented.) and dimension (Formula presented.). A class of infinite families of complete (Formula presented.) -arcs in (Formula presented.) is constructed, for (Formula presented.) a power of an odd prime (Formula presented.). The order of magnitude of (Formula presented.) is smaller than (Formula presented.). This property significantly distinguishes the complete (Formula presented.) -arcs of this paper from the previously known infinite families, whose size differs from (Formula presented.) by at most (Formula presented.)
On complete (N,d)-arcs derived from plane curves
AbstractIn this paper, we present several new complete (N,d)-arcs obtained from Fq-rational points of plane curves
On the Dickson-Guralnick-Zieve curve
The Dickson-Guralnick-Zieve curve, briefly DGZ curve, defined over the finite
field arises naturally from the classical Dickson invariant of
the projective linear group . The DGZ curve is an
(absolutely irreducible, singular) plane curve of degree and genus
In this paper we show that the DGZ curve has
several remarkable features, those appearing most interesting are: the DGZ
curve has a large automorphism group compared to its genus albeit its
Hasse-Witt invariant is positive; the Fermat curve of degree is a
quotient curve of the DGZ curve; among the plane curves with the same degree
and genus of the DGZ curve and defined over , the DGZ curve
is optimal with respect the number of its -rational points
Plane curves giving rise to blocking sets over finite fields
In recent years, many useful applications of the polynomial method have
emerged in finite geometry. Indeed, algebraic curves, especially those defined
by R\'edei-type polynomials, are powerful in studying blocking sets. In this
paper, we reverse the engine and study when blocking sets can arise from
rational points on plane curves over finite fields. We show that irreducible
curves of low degree cannot provide blocking sets and prove more refined
results for cubic and quartic curves. On the other hand, using tools from
number theory, we construct smooth plane curves defined over of
degree at most whose points form blocking sets.Comment: 25 page