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) and whose block are the subspaces of codimension two, where n ≥ 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 = 2 and obtains a Dembowski-Wagner type result for the class of all such quasi-symmetric designs.