Doubly transitive dimensional dual hyperovals: universal covers and non-bilinear examples

By [4] a doubly transitive, non-solvable dimensional dual hyperoval D is isomorphic either to the Mathieu dual hyperoval or to a quotient of a Huybrechts dual hyperoval. In order to determine all doubly transitive dimensional dual hyperovals, it remains to classify the solvable ones, 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, where E is elementary abelian of order 2n and S ≤ Γ L(1, 2n); see Yoshiara [12]. The known examples D are bilinear. In [1] the bilinear, doubly transitive, solvable dimensional dual hyperovals D of rank n with GL(1, 2n) ≤ S are 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 by [2, Cor. 1.3]. We determine the universal covers of the presently known doubly transitive dimensional dual hyperovals.

The non-solvable doubly transitive dimensional dual hyperovals

In [9] S. Yoshiara determines possible automorphism group of doubly transitive dimensional dual hyperovals. He shows that a doubly transitive dual hyperovalDis either isomorphic to the Mathieu dual hyperoval or the dual hyperoval is defined over 𝔽2, and if the hyperoval has rankn, the automorphism group has the formE⋅S, with an elementary abelian groupEof order 2nandSa subgroup of GL(n,2) acting transitively on the nontrivial elements ofE. Moreover Yoshiara describes the possible candidates forS. In this paper we assume thatSis non-solvable and show that then the dimensional dual hyperoval is a bilinear quotient of a Hyubrechts dual hyperoval.