Notes on Taniguchi’s dimensional dual hyperovals

The d -dimensional dual hyperoval T σ ( K ) constructed by Taniguchi is shown to be a quotient of the d -dimensional dual hyperoval H V d ( q ) constructed by exploiting the Veronesean map. We determine when the d -dimensional dual hyperoval S σ , ϕ d + 1 coincides with T σ ′ ( K ) . Some applications are also included.