63 research outputs found
Hermitian codes from higher degree places
Matthews and Michel investigated the minimum distances in certain
algebraic-geometry codes arising from a higher degree place . In terms of
the Weierstrass gap sequence at , they proved a bound that gives an
improvement on the designed minimum distance. In this paper, we consider those
of such codes which are constructed from the Hermitian function field. We
determine the Weierstrass gap sequence where is a degree 3 place,
and compute the Matthews and Michel bound with the corresponding improvement.
We show more improvements using a different approach based on geometry. We also
compare our results with the true values of the minimum distances of Hermitian
1-point codes, as well as with estimates due Xing and Chen
Group-labeled light dual multinets in the projective plane (with Appendix)
In this paper we investigate light dual multinets labeled by a finite group
in the projective plane defined over a field .
We present two classes of new examples. Moreover, under some conditions on the
characteristic , we classify group-labeled light dual multinets
with lines of length least
3-nets realizing a diassociative loop in a projective plane
A \textit{-net} of order is a finite incidence structure consisting of
points and three pairwise disjoint classes of lines, each of size , such
that every point incident with two lines from distinct classes is incident with
exactly one line from each of the three classes. The current interest around
-nets (embedded) in a projective plane , defined over a field
of characteristic , arose from algebraic geometry. It is not difficult to
find -nets in as far as . However, only a few infinite
families of -nets in are known to exist whenever , or .
Under this condition, the known families are characterized as the only -nets
in which can be coordinatized by a group. In this paper we deal with
-nets in which can be coordinatized by a diassociative loop
but not by a group. We prove two structural theorems on . As a corollary, if
is commutative then every non-trivial element of has the same order,
and has exponent or . We also discuss the existence problem for such
-nets
A characterization of the Artin-Mumford curve
Let be the Artin-Mumford curve over the finite prime field
with . By a result of Valentini and Madan,
\mbox{Aut}_{\mathbb{F}_p}(\mathcal{M})\cong H with . We prove that if is an algebraic curve of genus
such that \mbox{Aut}_{\mathbb{F}_p}(\mathcal{X}) contains a
subgroup isomorphic to then is birationally equivalent over
to the Artin-Mumford curve
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
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