Designs with the symmetric difference property on 64 points and their groups (Q1331141)

A symmetric 2-design has the symmetric difference property, or is an SDP design, if the symmetric difference of any three blocks is either a block or the complement of a block. The parameters \((v,k,\lambda)\) of a symmetric SDP design are of the form \[ v= 2^{2m},\quad k= 2^{2m-1}-2^{m-1},\quad \lambda= 2^{2m-2}-2^{m-1}. \] The symmetric SDP designs were characterized by \textit{J. F. Dillon} and \textit{J. R. Schatz} [Block designs with the symmetric difference property, Proc. NSA Math. Sci. Meet., 159-164 (1987)] as designs formed by the minimum weight vectors in a binary code spanned by a first-order Reed-Muller code and the incidence vector of a bent function. There are precisely four nonisomorphic SDP \(2\)-\((64,28,12)\) designs. In this paper the authors report the results of the computation of the automorphism groups of these four designs as well as the groups of their derived and residual designs. One interesting fact is that all four designs can be obtained from a difference set. As an application, the authors are able to enumerate the binary self-complementary \((28,7,12)\) and \((36,7,16)\) codes up to equivalence.