It is shown in Yoshiara (2004) that, if e4845dff9bb032121e41e88" title="Click to view the MathML source">d-dimensional dual hyperovals exist in V(n,2) (e4e3d0f3" title="Click to view the MathML source">GF(2)-vector space of rank e52ed78437bec" title="Click to view the MathML source">n), then a001ebfb7" title="Click to view the MathML source">2d+1≤n≤(d+1)(d+2)/2+2, and conjectured that n≤(d+1)(d+2)/2. Known bilinear dual hyperovals in a6d7" title="Click to view the MathML source">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 e4663bfbd68105409b7864a">, where the dual hyperovals e48e41d51234d4cb58d26"> and a6b8dc1fdb3a3a7c" title="Click to view the MathML source">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 e4845dff9bb032121e41e88" title="Click to view the MathML source">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 a6b8dc1fdb3a3a7c" title="Click to view the MathML source">Sc(l,GF(2r)), and show that a037b5db965767da9" title="Click to view the MathML source">Aut(Sc(l2,GF(2rl1)))<Aut(Sc(l,GF(2r))).