On some bilinear dual hyperovals

Abstract

It is shown in Yoshiara (2004) that, if d-dimensional dual hyperovals exist in V(n,2) (GF(2)-vector space of rank n), then 2d+1≤n≤(d+1)(d+2)/2+2, and conjectured that n≤(d+1)(d+2)/2. Known bilinear dual hyperovals in V((d+1)(d+2)/2,2) are the Huybrechts dual hyperoval and the Buratti–Del Fra dual hyperoval. In this paper, we investigate on the covering map π:Sc′(l′,GF(2r′))→Sc(l,GF(2r)), where the dual hyperovals Sc′(l′,GF(2r′)) and Sc(l,GF(2r)) are constructed in Taniguchi (2014). Using the result, we show that the Buratti–Del Fra dual hyperoval has a bilinear quotient in V(2d+1,2) if d is odd. On the other hand, we show that the Huybrechts dual hyperoval has no bilinear quotient in V(2d+1,2). We also determine the automorphism group of Sc(l,GF(2r)), and show that Aut(Sc(l2,GF(2rl1)))<Aut(Sc(l,GF(2r))).