On d-dual hyperovals in

In P G ( d ( d + 3 ) / 2 , 2 ) , there are three known families of dual hyperovals; Huybrechtsʹs dual hyperovals ([C. Huybrechts, Dimensional dual hyperovals in projective spaces and c.AC* geometries, Discrete math. 255 (2002), 503–532]), Veronesean dual hyperovals ([J. Thas and H. van Maldeghem, Characterizations of the finite quadric Veroneseans V n 2 n , Quart. J. Math. Oxford. 55 (2004), 99–113] and [S. Yoshiara, Ambient spaces of dimensional dual arcs, J. Alg. Combin. 19 (2004), 5–23]), and the family constructed by Buratti and Del Fra ([M. Buratti and A. del Fra, Semi-Boolean quadruple systems and dimensional dual hyperovals, Adv. Geom. 3 (2003), 245–253] and [A. del Fra and S. Yoshiara, Dimensional dual hyperovals associated with Steiner systems, Europ. J. Combinatorics. 26 (2005), 173–194]) based on Huybrechtsʹs dual hyper-ovals. In this note, we construct a family of dual hyperovals based on Veronesean dual hyperovals. We also prove that our dual hyperovals are not isomorphic to the Veronesean dual hyperovals and do not satisfy Property (T ). Hence, we have a new family of dual hyperovals in P G ( d ( d + 3 ) / 2 , 2 ) .