Regular Triples with Respect to a Hyperoval

In a previous paper, [6], we associated with every hyperoval of a projective plane of even order a Hadamard \(2\)–design and investigated when this design has lines with three points. We study further this problem using the concept of regular triple and prove the existence of lines with three points in Hadamard designs associated with translation hyperovals. In the general case, the existence of a secant line of regular triples implies that the order of the projective plane is a power of two.