The dual Yoshiara construction gives new extended generalized quadrangles

Abstract

A Yoshiara family is a set of q+3 planes in PG(5,q),q even, such that for any element of the set the intersection with the remaining q+2 elements forms a hyperoval. In 1998 Yoshiara showed that such a family gives rise to an extended generalized quadrangle of order (q+1,q−1). He also constructed such a family S(〇) from a hyperoval 〇 in PG(2,q). In 2000 Ng and Wild showed that the dual of a Yoshiara family is also a Yoshiara family. They showed that if 〇 has o-polynomial a monomial and 〇 is not regular, then the dual of S(〇) is a new Yoshiara family. This article extends this result and shows that in general the dual of S(〇) is a new Yoshiara family, thus giving new extended generalized quadrangles.