On \(d\)-dimensional dual hyperovals in PG\((2d,2)\) (Q993650)

A family \(S\) of \(2^{d+1}\) \(d\)-dimensional subspaces of \(PG(m,2)\) is called a \textit{\(d\)-dimensional dual hyperoval} if any two distinct members of \(S\) intersect in precisely one point, no three members of \(S\) pass through a common point, and the union of the members of \(S\) generates all of \(PG(m,2)\). In the paper under review \(d\)-dimensional dual hyperovals in \(PG(2d,2)\) are constructed by using spreads associated with regular nearfield planes of even characteristic whose dimension over their kernel is greater than \(2\). It is shown that these dual hyperovals are not isomorphic to the ones constructed by \textit{S. Yoshiara} [Eur. J. Comb. 20, No.~6, 589--603 (1999; Zbl 0937.51009)], and hence there exist non-isomorphic \(d\)-dimensional dual hyperovals of \(PG(2d,2)\) for \(d>2\). It should be noted that this cannot happen when \(d=2\), as shown in [\textit{A. Del Fra}, Geom. Dedicata 79, No.~2, 157--178 (2000; Zbl 0948.51008)].