  Many generator matrices for constructing extremal binary self-dual codes of
different lengths have the form G=(I|A), where I is the n by n identity matrix
and A is the n by n matrix fully determined by the first row. In this work, we
define a generator matrix in which A is a block matrix, where the blocks come
from group rings and also, A is not fully determined by the elements appearing
in the first row. By applying our construction over F_2+uF_2 and by employing
the extension method for codes, we were able to construct new extremal binary
self-dual codes of length 68. Additionally, by employing a generalised
neighbour method to the codes obtained, we were able to construct many new
binary self-dual [68,34,12]-codes with the rare parameters gamma=7,8 and 9 in
W_{68,2}. In particular, we find 92 new binary self-dual [68,34,12]-codes.
