Designs with the symmetric difference property on 64 points and their groups

Abstract

The automorphism groups of the symmetric 2-(64, 28, 12) designs with the symmetric difference property (SDP), as well as the groups of their derived and residual designs, are computed. The symmetric SDP designs all have transitive automorphism groups. In addition, they all admit transitive regular subgroups, or equivalently, (64, 28, 12) difference sets. These results are used for the enumeration of certain binary codes achieving the Grey-Rankin bound and point sets of elliptic or hyperbolic type in PG(5, 2).