Quotients of the deformation of Veronesean dual hyperoval in PG(3d,2)PG(3d,2)

Let d≥3d≥3. In PG(d(d+3)/2,2)PG(d(d+3)/2,2), there are four known non-isomorphic dd-dimensional dual hyperovals by now. These are Huybrechts’ dual hyperoval by Huybrechts (2002) [4], Buratti-Del Fra’s dual hyperoval by Buratti and Del Fra (2003) [1], Del Fra and Yoshiara (2005) [3], Veronesean dual hyperoval by Thas and van Maldeghem (2004) [9], Yoshiara (2004) [12] and the dual hyperoval, which is a deformation of Veronesean dual hyperoval by Taniguchi (2009) [6].In this paper, using a generator σσ of the Galois group Gal(GF(2dm)/GF(2))Gal(GF(2dm)/GF(2)) for some m≥3m≥3, we construct a dd-dimensional dual hyperoval TσTσ in PG(3d,2)PG(3d,2), which is a quotient of the dual hyperoval of [6]. Moreover, for generators σ,τ∈Gal(GF(2dm)/GF(2))σ,τ∈Gal(GF(2dm)/GF(2)), if TσTσ and TτTτ are isomorphic, then we show that σ=τσ=τ or σ=τ−1σ=τ−1 on GF(2d)GF(2d). Hence, we see that there are many non-isomorphic quotients in PG(3d,2)PG(3d,2) for the dual hyperoval of [6] if dd is large.