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

Abstract

AbstractBy [4] a doubly transitive, non-solvable dimensional dual hyperovalDis 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 hyperovalDof ranknis defined over 𝔽2and has an automorphism of the formES, whereEis elementary abelian of order 2nandS≤ Γ L(1, 2n); see Yoshiara [12]. The known examplesDare bilinear. In [1] thebilinear, doubly transitive, solvable dimensional dual hyperovalsDof ranknwith GL(1, 2n) ≤Sare classified. Here we present two new classes ofnon-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.