Exponential number of quasi-symmetric SDP designs and codes meeting the Grey-Rankin bound

It is shown that quasi-symmetric designs which are derived or residual designs of nonisomorphic symmetric designs with the symmetric difference property are also nonisomorphic. Combined with a result by W. Kantor, this implies that the number of nonisomorphic quasi-symmetric designs with the symmetric difference property grows exponentially. The column spaces of the incidence matrices of these designs provide an exponential number of inequivalent codes meeting the Grey-Rankin bound. A transformation of quasi-symmetric designs by means of maximal arcs is described. In particular, a residual quasi-symmetric design with the symmetric difference property is transformed into a quasi-symmetric design with the same block graph but higher rank over GF(2).Dedicated to Professor Hanfried Lenz on the occasion of his 75th birthday.This paper was written while the author was at the University of Giessen as a Research Fellow of the Alexander von Humboldt Foundation, on leave from the University of Sofia, Bulgaria.