A new construction of the -dimensional Buratti–Del Fra dual hyperoval

The Buratti–Del Fra dual hyperoval D d ( F 2 ) is one of the four known infinite families of simply connected d -dimensional dual hyperovals over F 2 with ambient space of vector dimension ( d + 1 ) ( d + 2 ) / 2  (Buratti and Del Fra (2003) [1]). A criterion (Proposition 1) is given for a d -dimensional dual hyperoval over F 2 to be covered by D d ( F 2 ) in terms of the addition formula. Using it, we provide a simpler model of D d ( F 2 )  (Proposition 3). We also give conditions (Lemma 4) for a collection S [ B ] of ( d + 1 ) -dimensional subspaces of K ⊕ K constructed from a symmetric bilinear form B on K ≅ F 2 d + 1 to be a quotient of D d ( F 2 ) . For when d is even, an explicit form B satisfying these conditions is given. We also provide a proof for the fact that the affine expansion of D d ( F 2 ) is covered by the halved hypercube (Proposition 10).