A characterization of projective subspaces of codimension two as quasi-symmetric designs with good blocks

Consider an incidence structure whose points are the points of a PGn(n+2,q)PGn(n+2,q) and whose block are the subspaces of codimension two, where n⩾2n⩾2. Since every two subspaces of codimension two intersect in a subspace of codimension three or codimension four, it is easily seen that this incidence structure is a quasi-symmetric design. The aim of this paper is to prove a characterization of such designs (that are constructed using projective geometries) among the class of all the quasi-symmetric designs with correct parameters and with every block a good block. The paper also improves an earlier result for the special case of n=2n=2 and obtains a Dembowski–Wagner-type result for the class of all such quasi-symmetric designs.