Semi-Boolean Steiner systems and dimensional dual hyperovals

A dimensional dual hyperoval satisfying property (H) [61 in a projective space of order 2 is naturally associated with a "semi-Boolean" Steiner quadruple system. The only known examples are associated with Boolean systems. For every d > 2, we construct a new d-dimensional dual hyperoval satisfying property (H) in PG(d(d + 3)/2,2); its related semi-Boolean system is the Teirlinck one. It is universal and admits quotients in PG(n, 2), with 4d < n < d(d + 3)/2, if d greater than or equal to 6. We also prove the uniqueness of d-dimensional dual hyperovals satisfying property (H) in PG(d(d + 3)/2,2), whose related semi-Boolean systems belongs to a particular class, which includes Boolean and Teirlinck systems. Finally, we prove property (mI) [6] for them.