17 research outputs found
On AG codes from a generalization of the Deligne-Lustzig curve of Suzuki type
In this paper, Algebraic-Geometric (AG) codes and quantum codes associated to
a family of curves which comprises the famous Suzuki curve are investigated.
The Weierstrass semigroup at some rational point is computed. Notably, each
curve in the family turn out to be a Castle curve over some finite field, and a
weak Castle curve over its extensions. This is a relevant feature when codes
constructed from the curve are considered
Curves with more than one inner Galois point
Let be an irreducible plane curve of
where is an algebraically closed field of characteristic . A point is an inner Galois point for if
the projection from is Galois. Assume that has two
different inner Galois points and , both simple. Let and
be the respective Galois groups. Under the assumption that fixes ,
for , we provide a complete classification of and we exhibit a curve for each such . Our proof relies on deeper
results from group theory
Complete -arcs in from the Hermitian curve
We prove that, if is large enough, the set of the
-rational points of the Hermitian curve is a complete
-arc in , addressing an open case from
a recent paper by Korchm\'aros, Sz\H{o}nyi and Nagy. An algebraic approach
based on the investigation of some algebraic varieties attached to the arc is
used
Minimal codewords in Norm-Trace codes
In this paper, we consider the affine variety codes obtained evaluating the
polynomials , , at the
affine \F_{q^r}-rational points of the Norm-Trace curve. In particular, we
investigate the weight distribution and the set of minimal codewords. Our
approach, which uses tools of algebraic geometry, is based on the study of the
absolutely irreducibility of certain algebraic varieties
New sextics of genus 6 and 10 attaining the Serre bound
We provide new examples of curves of genus 6 or 10 attaining the Serre bound.
They all belong to the family of sextics introduced in [19] as a a
generalization of the Wiman sextics [36] and Edge sextics [9]. Our approach is
based on a theorem by Kani and Rosen which allows, under certain assumptions,
to fully decompose the Jacobian of the curve. With our investigation we are
able to update several entries in \url{http://www.manypoints.org} ([35])
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