A construction of 2-designs from Steiner systems and extendable designs

We give a construction of a 2- ( m n 2 + 1 , m n , ( n + 1 ) ( m n − 1 ) ) design starting from a Steiner system S ( 2 , m + 1 , m n 2 + 1 ) and an affine plane of order n. This construction is applied to known classes of Steiner systems arising from affine and projective geometries, Denniston designs, and unitals. We also consider the extendability of these designs to 3-designs.