On symmetric (78,22,6) designs and related self-orthogonal codes

We describe a method for constructing 2-designs admitting a solvable automorphism group, and construct new symmetric (78, 22, 6) designs. Until now, only five symmetric (78, 22, 6) designs were known. In this paper we show that, up to isomorphism, there are at least 413 symmetric (78, 22, 6) designs, 412 of them having Z(6) as an automorphism group. Further, we show that up to isomorphism there is exactly one symmetric (78, 22, 6) design admitting an automorphism group isomorphic to Frob(39) x Z(2), namely the design constructed by Zvonimir Janko and Tran van Trung. Thus, there is no (78,22,6) difference set in the group Frob(39) x Z(2). We study binary linear codes spanned by the incidence matrices of the constructed designs. Further, extending previous results on codes obtained from orbit matrices of 2-designs, we show that under certain conditions both fixed and nonfixed part of an orbit matrix span a self-orthogonal code over the finite field F-p(n) or over the ring Z(m). We construct self-orthogonal codes over Z(4) spanned by orbit matrices of symmetric (78, 22, 6) designs.