Notes on Taniguchi’s dimensional dual hyperovals

The dd-dimensional dual hyperoval Tσ(K)Tσ(K) constructed by Taniguchi is shown to be a quotient of the dd-dimensional dual hyperoval HVd(q)HVd(q) constructed by exploiting the Veronesean map. We determine when the dd-dimensional dual hyperoval Sσ,ϕd+1 coincides with Tσ′(K)Tσ′(K). Some applications are also included.