46 research outputs found
An infinite family of hyperovals of , even
We construct an infinite family of hyperovals on the Klein quadric
, even. The construction makes use of ovoids of the symplectic
generalized quadrangle that is associated with an elliptic quadric which
arises as solid intersection with . We also solve the isomorphism
problem: we determine necessary and sufficient conditions for two hyperovals
arising from the construction to be isomorphic
On the dual of the dual hyperoval from APN function f(x)=x3+Tr(x9)
AbstractUsing a quadratic APN function f on GF(2d+1), Yoshiara (2009) [15] constructed a d-dimensional dual hyperoval Sf in PG(2d+1,2). In Taniguchi and Yoshiara (2005) [13], we prove that the dual of Sf, which we denote by Sf⊥, is also a d-dimensional dual hyperoval if and only if d is even. In this note, for a quadratic APN function f(x)=x3+Tr(x9) on GF(2d+1) by Budaghyan, Carlet and Leander (2009) [2], we show that the dual Sf⊥ and the transpose of the dual Sf⊥T are not isomorphic to the known bilinear dual hyperovals if d is even and d⩾6
Abstract hyperovals, partial geometries, and transitive hyperovals
Includes bibliographical references.2015 Summer.A hyperoval is a (q+2)- arc of a projective plane π, of order q with q even. Let G denote the collineation group of π containing a hyperoval Ω. We say that Ω is transitive if for any pair of points x, y is an element of Ω, there exists a g is an element of G fixing Ω setwise such that xg = y. In1987, Billotti and Korchmaros proved that if 4||G|, then either Ω is the regular hyperoval in PG(2,q) for q=2 or 4 or q = 16 and |G||144. In 2005, Sonnino proved that if |G| = 144, then π is desarguesian and Ω is isomorphic to the Lunelli-Sce hyperoval. For our main result, we show that if G is the collineation group of a projective plane containing a transitivehyperoval with 4 ||G|, then |G| = 144 and Ω is isomorphic to the Lunelli-Sce hyperoval. We also show that if A(X) is an abstract hyperoval of order n ≡ 2(mod 4); then |Aut(A(X))| is odd. If A(X) is an abstract hyperoval of order n such that Aut(A(X)) contains two distinct involutions with |FixX(g)| and |FixX(ƒ)| ≥ 4. Then we show that FixX(g) ≠FixX(ƒ). We also show that there is no hyperoval of order 12 admitting a group whose order is divisible by 11 or 13, by showing that there is no partial geometry pg(6, 10, 5) admitting a group of order 11 or of order 13. Finally, we were able to show that there is no hyperoval in a projective plane of order 12 with a dihedral subgroup of order 14, by showing that that there is no partial geometry pg(7, 12, 6) admitting a dihedral group of order 14. The latter results are achieved by studying abstract hyperovals and their symmetries
On Translation Hyperovals in Semifield Planes
In this paper we demonstrate the first example of a finite translation plane
which does not contain a translation hyperoval, disproving a conjecture of
Cherowitzo. The counterexample is a semifield plane, specifically a Generalised
Twisted Field plane, of order . We also relate this non-existence to the
covering radius of two associated rank-metric codes, and the non-existence of
scattered subspaces of maximum dimension with respect to the associated spread
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