Doubly Transitive Dimensional Dual Hyperovals: Universal Covers and Non-Bilinear Examples

In [5] we showed, that a doubly transitive, non-solvable dimensional dual hyperoval D is either isomorphic to the Mathieu dual hyperoval or to a quotient of a Huybrechts dual hyperoval. In order to determine the doubly transitive dimensional dual hyperovals, it remains to classify the doubly transitive, solvable dimensional dual hyperovals and this paper is a contribution to this problem. A doubly transitive, solvable dimensional dual hyperoval D of rank n is defined over F2 and has an automorphism of the form ES, E elementary abelian of order 2 and S ≤ ΓL(1, 2) (see Yoshiara [13]). The known doubly transitive, solvable dimensional dual hyperovals D are bilinear. In [2] the bilinear, doubly transitive, solvable dimensional dual hyperovals D of rank n with GL(1, 2) ≤ S have been classified. Here we present two new classes of non-bilinear, doubly transitive dimensional dual hyperovals. We also consider universal covers of doubly transitive dimensional dual hyperovals, since they are again doubly transitive dimensional dual hyperovals (see [3, Cor. 1.3]). We shall determine the universal covers of the presently known doubly transitive dimensional dual hyperovals.