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

Abstract

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).