On dimensional dual hyperovals 𝒮σ,ϕd+1
Hiroaki Taniguchi, Satoshi Yoshiara · Innovations in incidence geometry · 2005
A d-dimensional dual hyperoval S d+1 σ,φ inside P G(2d + 1, 2) (d ≥ 2) is constructed in [5], for a generator σ of Gal(GF (q)/GF (2)) and an o-polynomial φ(X) of GF (q)[X] (q = 2 d+1 ).There, its automorphism group is determined and a criterion is given for these dimensional dual hyperovals to be isomorphic, assuming that the map φ on GF (q) induced by φ(X) lies in Gal(GF (q)/GF (2)).In this paper, we extend these results for a monomial o-polynomial φ.We show that Aut(σ,φ is always fixed by any automorphism of S d+1 σ,φ .Furthermore, S d+1 σ,φ ∼ = S d+1 σ ,φ if and only if either (σ, φ) = (σ , φ ) or σσ = φφ = id.