  We introduce an altered version of the four circulant construction over group
rings for self-dual codes. We consider this construction over the binary field,
the rings F_2 + uF_2 and F_4 + uF_4; using groups of order 3, 7, 9, 13, and 15.
Through these constructions and their extensions, we find binary self-dual
codes of lengths 32, 40, 56, 64, 68 and 80, all of which are extremal or
optimal. In particular, we find five new self-dual codes of parameters [56, 28,
10], twenty-three extremal binary self-dual codes of length 68 with new weight
enumerators and fifteen new self-dual codes of parameters [80, 40, 14].
