Projective Paley sets

A two-character set in PG(r,q) is a set X of points with the property that the intersection number with any hyperplane only takes two values. A projective Paley set of PG(2n-1,q),q odd, is a subset X of points such that every hyperplane of PG(2n-1,q) meets X in either (qn+1)(qn-1-1)/2(q-1) or (qn-1)(qn-1+1)/2(q-1) points. A quasi-quadric in PG(2n-1,q) is a two-character set that has the same size and the same intersection numbers with respect to hyperplanes as a nondegenerate quadric. Here we construct projective Paley sets of PG(3,q) left invariant by a cyclic group of order q2+1 and of PG(5,q) admitting PSL(2,q2) as an automorphism group. Also infinite families of quasi-quadrics of PG(5,q) are provided.