On dd-dimensional Buratti–Del Fra type dual hyperovals 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 (Huybrechts (2002) [6]), Buratti–Del Fra’s dual hyperoval (Buratti and Del Fra (2003) [1], Del Fra and Yoshiara (2005) [3]), Veronesean dual hyperoval (Thas and van Maldeghem (2004) [9], Yoshiara (2004) [12]), and the dual hyperoval which is a deformation of Veronesean dual hyperoval (Taniguchi (2009) [8]). Using quadratic APN functions on GF(2d+1)GF(2d+1), Yoshiara (2009) [10], constructed dd-dimensional dual hyperovals in PG(2d+1,2)PG(2d+1,2). This construction enables us to investigate quadratic APN functions from the view point of dual hyperovals (Edel (2009) [4]). Yoshiara generalized this construction in Yoshiara (2008) [11]. Note that these Yoshiara’s dual hyperovals are quotients of Huybrechts’ dual hyperoval. In this note, using quadratic APN functions on GF(2d)GF(2d), we construct dd-dimensional dual hyperovals in PG(3d,2)PG(3d,2), which are quotients of Buratti–Del Fra’s dual hyperoval. Moreover, we prove that, if two of these dual hyperovals are isomorphic, then the corresponding quadratic APN functions are extended affine equivalent. We call these dual hyperovals Buratti–Del Fra type dual hyperovals. Then, if dd is sufficiently large, there are many non-isomorphic dd-dimensional Buratti–Del Fra type dual hyperovals in PG(3d,2)PG(3d,2).