Geometric constructions of two-character sets

A two-character set in a finite projective space is a set of points with the property that the intersection number with any hyperplanes only takes two values. In this paper constructions of some two-character sets are given. In particular, infinite families of tight sets of the symplectic generalized quadrangle W(3,q2) and the Hermitian surface H(3,q2) are provided. A quasi-Hermitian variety H in PG(r,q2) is a combinatorial generalization of the (non-degenerate) Hermitian variety H(r,q2) so that H and H(r,q2) have the same number of points and the same intersection numbers with hyperplanes. Here we construct two families of quasi-Hermitian varieties, for r,q both odd, admitting PΓO+ (r+1,q) and PΓO- (r+1,q) as automorphisms group.