122 research outputs found
A geometric approach to Mathon maximal arcs
In 1969 Denniston gave a construction of maximal arcs of degree d in
Desarguesian projective planes of even order q, for all d dividing q. In 2002
Mathon gave a construction method generalizing the one of Denniston. We will
give a new geometric approach to these maximal arcs. This will allow us to
count the number of isomorphism classes of Mathon maximal arcs of degree 8 in
PG(2,2^h), h prime.Comment: 20 page
On Mathon's construction of maximal arcs in Desarguesian planes. II
In a recent paper [M], Mathon gives a new construction of maximal arcs which
generalizes the construction of Denniston. In relation to this construction,
Mathon asks the question of determining the largest degree of a non-Denniston
maximal arc arising from his new construction. In this paper, we give a nearly
complete answer to this problem. Specifically, we prove that when and
, the largest of a non-Denniston maximal arc of degree in
PG(2,2^m) generated by a {p,1}-map is (\floor {m/2} +1). This confirms our
conjecture in [FLX]. For {p,q}-maps, we prove that if and ,
then the largest of a non-Denniston maximal arc of degree in
PG(2,2^m) generated by a {p,q}-map is either \floor {m/2} +1 or \floor{m/2}
+2.Comment: 21 page
Combinatorial problems in finite geometry and lacunary polynomials
We describe some combinatorial problems in finite projective planes and
indicate how R\'edei's theory of lacunary polynomials can be applied to them
On the structure of the directions not determined by a large affine point set
Given a point set in an -dimensional affine space of size
, we obtain information on the structure of the set of
directions that are not determined by , and we describe an application in
the theory of partial ovoids of certain partial geometries
Direction problems in affine spaces
This paper is a survey paper on old and recent results on direction problems
in finite dimensional affine spaces over a finite field.Comment: Academy Contact Forum "Galois geometries and applications", October
5, 2012, Brussels, Belgiu
- …