On \(d\)-dimensional Buratti-Del Fra type dual hyperovals in \(PG(3d,2)\)
Given an \(m\)-dimensional projective space over \(GF(2)\), \(PG(m, 2)\), and an integer \(d\), where \(m > d \geq 2\), a set \(S\) of \(d\)-dimensional subspaces of \(PG(m, 2)\) is called a \(d\)-dimensional \textit{dual hyperoval} in \(PG(m, 2)\) if: 1) any two distinct elements of \(S\) share a point, 2) any three different elements of \(S\) have a trivial intersection, 3) the union of the elements of \(S\) generates the whole \(PG(m, 2)\) and 4) the cardinality of \(S\) is \(2^{d+1}\); the projective space \(PG(m, 2)\) is called the ambient space of \(S\). Two distinct dual hyperovals \(S_1\) and \(S_2\) that have the same ambient space are isomorphic if there exists a linear automorphism of the ambient space sending the elements of \(S_1\) onto the elements of \(S_2\). For \(d \geq 3\), four non-isomorphic \(d\)-dimensional dual hyperovals are so far known to exist in the ambient space \(PG(d(d+3)/2, 2)\), namely: Huybrechts' dual hyperoval, Buratti-Del Fra's dual hyperoval, Veronesean dual hyperoval and the dual hyperoval that is a deformation of the last-mentioned one. In the present paper, the author -- following and extending the strategy first put forward by S. Yoshiara -- employs a quadratic APN (almost perfect non-linear) function \(f\) on \(GF(2^d)\) and constructs a \(d\)-dimensional dual hyperoval \(D_f\) in \(PG(3d, 2)\) that is a quotient of Buratti-Del Fra's dual hyperoval. Then he proves that given two different quadratic APN functions \(f\) and \(g\) on \(GF(2^d)\), if \(D_f\) is isomorphic to \(D_g\) then \(f\) and \(g\) are extended affine \textit{equivalent} functions on \(GF(2^d)\). The proofs are rather lengthy and entail a lot of technicalities.
scientific article; zbMATH DE number 5817556