  The $\Z_{2^s}$-additive codes are subgroups of $\Z^n_{2^s}$, and can be seen
as a generalization of linear codes over $\Z_2$ and $\Z_4$. A $\Z_{2^s}$-linear
code is a binary code which is the Gray map image of a $\Z_{2^s}$-additive
code. We consider $\Z_{2^s}$-additive simplex codes of type $\alpha$ and
$\beta$, which are a generalization over $\Z_{2^s}$ of the binary simplex
codes. These $\Z_{2^s}$-additive simplex codes are related to the
$\Z_{2^s}$-additive Hadamard codes. In this paper, we use this relationship to
establish the kernel of their binary images, under the Gray map, the
$\Z_{2^s}$-linear simplex codes. Similar results can be obtained for the binary
Gray map image of $\Z_{2^s}$-additive MacDonald codes.
