  In this work, we study codes over the ring
R_{k,m}=F_2[u,v]/<u^{k},v^{m},uv-vu>, which is a family of Frobenius,
characteristic 2 extensions of the binary field. We introduce a distance and
duality preserving Gray map from R_{k,m} to F_2^{km} together with a Lee
weight. After proving the MacWilliams identities for codes over R_{k,m} for all
the relevant weight enumerators, we construct many binary self-dual codes as
the Gray images of self-dual codes over R_{k,m}. In addition to many extremal
binary self-dual codes obtained in this way, including a new construction for
the extended binary Golay code, we find 175 new Type I binary self-dual codes
of parameters [72,36,12] and 105 new Type II binary self-dual codes of
parameter [72,36,12].
