Characterization of Certain Minimal Rank Designs

Dillon asked whether the all-1 vector is in the binary code of a square (symmetric) design with parameters (22m, 22m−1−2m−1, 22m−2−2m−1) and dimension 2m+2. In this paper we show that the answer to this question is yes. This result gives a characterization of designs with these parameters and minimal 2-rank as SDP designs. Our result also allows us to remove a hypothesis from a theorem of Dillon and Schatz relating difference sets in elementary abelian 2-groups to SDP designs. Along the way we prove results about any designs with the parameters of the residual and derived designs. One of the results deals with a divisibility property that characterizes the elliptic and hyperbolic designs.