**Abstarct **
In an earlier paper the authors have studied simplex codes of type
$\alpha$ and $\beta$ over $\Z_4$ and obtained some known binary linear
and nonlinear codes as Gray images of these codes. In this correspondence,
we give the Hamming, Lee, homogeneous weight distributions, and the generalized Hamming weights
of simplex codes of type $\alpha$ and $\beta$ over $\Z_{2^{s}}.$ The Generalized Gray
map is then used to construct some binary codes; among them a few are linear and
meet the Griesmer bound. We also give the weight hierarchies of the first order
Reed-Muller codes over $\Z_{2^{s}}.$ The above codes are also shown to
satisfy the chain condition.