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

The Buratti–Del Fra dual hyperoval Dd(F2) is one of the four known infinite families of simply connected dd-dimensional dual hyperovals over F2 with ambient space of vector dimension (d+1)(d+2)/2(d+1)(d+2)/2 (Buratti and Del Fra (2003) [1]). A criterion (Proposition 1) is given for a dd-dimensional dual hyperoval over F2 to be covered by Dd(F2) in terms of the addition formula. Using it, we provide a simpler model of Dd(F2) (Proposition 3). We also give conditions (Lemma 4) for a collection S[B]S[B] of (d+1)(d+1)-dimensional subspaces of K⊕KK⊕K constructed from a symmetric bilinear form BB on K≅F2d+1 to be a quotient of Dd(F2). For when dd is even, an explicit form BB satisfying these conditions is given. We also provide a proof for the fact that the affine expansion of Dd(F2) is covered by the halved hypercube (Proposition 10).

► We provide a new construction of the dd-dimensional Buratti–Del Fra dual hyperoval.
► A criterion for being covered by the Buratti–Del Fra dimensional dual hyperoval is given.
► We construct a proper quotient of the dd-dimensional Buratti–Del Fra dual hyperoval.
► The halved hypercube covers the affine expansion of the Buratti–Del Fra dual hyperoval.