  In this work, we define three composite matrices derived from group rings. We
employ these composite matrices to create generator matrices of the form [In |
{\Omega}(v)], where In is the identity matrix and {\Omega}(v) is a composite
matrix and search for binary self-dual codes with parameters [36, 18, 6 or 8].
We next lift these codes over the ring R1 = F2 + uF2 to obtain codes whose
binary images are self-dual codes with parameters [72,36,12]. Many of these
codes turn out to have weight enumerators with parameters that were not known
in the literature before. In particular, we find 30 new Type I binary self-dual
codes with parameters [72, 36, 12].
