[Turkmath:962] Workshop On Finite Fields: Arcs, Curves and Bent Functions (Sabanci University)

[Canan Kasikci (Alumni)]

Degerli liste uyeleri,

1-3 Subat arasinda Sabanci Universitesinde duzenlenecek olan ve asagida
detaylarini bulacaginiz etkinlige katiliminiz bizi mutlu edecektir.

Iyi gunler dilegiyle,

canan kasikci

Ulasim icin:
http://www.sabanciuniv.edu/en/transportation/shuttle-hours

*Workshop on Finite Fields: Arcs, Curves and Bent Functions*


*Program:*

*February 1st, Monday, (FENS G032)*
*13:35-14:30: *Simeon Ball, Extending small arcs to large arcs
*February 3rd, Wednesday, (FENS 2008)*
*11:00-12:00: *Nurdagul Anbar, A new tower meeting Zink's bound with good
p-rank
*13:30-14:30: *Wilfried Meidl, Bent, gbent and bent4 functions

*------------------------------**------------------------------*
*------------------------------*

*Abstracts:*

*Extending small arcs to large arcs*
*by Simeon Ball - Universitat Politècnica Catalunya*

Let Fq denote the finite field with q elements. An arc is a set of vectors
of the k-dimensional vector space over the finite field with q elements Fq,
in which every subset of size k is a basis of the space, i.e. every
k-subset is a set of linearly independent vectors. Alternatively (and
equivalently) one can define an arc as a subset of points in the (k
−1)-dimensional projective space over Fq, in which every subset of size k
spans the whole space. The matrix whose columns are the vectors of an arc S
generates a k-dimensional linear maximum distance separable (MDS) code over
Fq of length |S|, so arcs and linear MDS codes are equivalent objects. If a
small arc can be extended to a large arc and the characteristic is odd then
we prove that certain necessary conditions must be satisfied. For example,
it follows from Segre’s theorem that if the characteristic is odd and an
arc of size six can be extended to an arc of size q + 1 then it is
necessary that the small arc is contained in a conic. We prove further
results of this type. These theorems may provide new tools in the
computational classification and construction of large arcs.
*------------------------------**------------------------------*
*------------------------------*

*A new tower meeting Zink's bound with good p-rank*
*by Nurdagul Anbar, Denmark Technical University *

In a recent work [1], Bassa, Beelen, Garcia and Stichtenoth have introduced
a new recursive tower defined over Fqn for any n ≥ 2, which results in the
best known lower bound for Ihara's constant for any non-prime finite
fields.


In this work, we have investigated a tower F=Fq3 =(F1 ⊂ F2 ⊂.......)arising
from the Drinfeld modular interpretation of the tower in [1]. More
precisely, we have shown that the limit of F=Fq3 attains Zink's bound and
that the p-torsion limit of the tower F=Fq3 satisfies


[image: Inline image 2]




where γ (Fn) and g(Fn) is the p-rank and the genus of Fn, respectively.
Moreover, we have shown that the equality holds in (1) in the case of prime
q. This is a joint work with Peter Beelen and Nhut Nguyen.

*References:*[1] A. Bassa, P. Beelen, A. Garcia, H. Stichtenoth, Towers of
Function Fields over Non-prime Finite Fields, Moscow Mathematical Journal 15
(1) (2015) 1{29.

*-----------------------------------------------------------------------------------------*


*Bent, gbent and bent4 functions*

*by Wilfried Meidl, RICAM Linz*



After recalling bent functions, vectorial bent functions and their relations
to difference sets, two types of generalizations of bent functions are
discussed.
The first one are gbent functions (generalized bent functions), which are
functions from F2 n   to Z2k  with a flat spectrum with respect to a
generalized
Walsh transform. Recent results on their structure, particularly on their
relation with (vectorial) bent and semibent functions will be presented.
The second are bent4 functions which appear as component functions of the
recently discovered modified planar functions, i.e., functions describing (2
n; 2n; 2n; 1)-relative difference sets and give rise to projective planes.
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