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(O) from a hyperoval O 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 O has o-polynomial a monomial and O is not regular, then the dual of S(O) is a new Yoshiara family. This article extends this result and shows that in general the dual of S(O) is a new Yoshiara family, thus giving new extended generalized quadrangles.
[1]
Siaw-Lynn Ng,et al.
A New Family of Extended Generalized Quadrangles of Order (q + 1, q - 1)
,
2000,
Eur. J. Comb..
[2]
Joseph A. Thas.
Some new classes of extended generalized quadrangles of order {$(q+1,q-1)$}
,
1998
.
[3]
Satoshi Yoshiara.
A Construction of Extended Generalized Quadrangles Using the Veronesean
,
1997,
Eur. J. Comb..
[4]
Satoshi Yoshiara.
The Universal Covers of a Family of Extended Generalized Quadrangles
,
1998,
Eur. J. Comb..
[5]
J. Thas,et al.
General Galois geometries
,
1992
.