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A new construction of the d-dimensional BurattiāDel Fra dual hyperoval
AbstractThe BurattiāDel Fra dual hyperoval Dd(F2) is one of the four known infinite families of simply connected d-dimensional dual hyperovals over F2 with ambient space of vector dimension (d+1)(d+2)/2Ā (Buratti and Del Fra (2003)Ā [1]). A criterionĀ (PropositionĀ 1) is given for a d-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)Ā (PropositionĀ 3). We also give conditionsĀ (LemmaĀ 4) for a collection S[B] of (d+1)-dimensional subspaces of KāK constructed from a symmetric bilinear form B on Kā
F2d+1 to be a quotient of Dd(F2). 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 Dd(F2) is covered by the halved hypercubeĀ (PropositionĀ 10)