On the Kernel of ${\mathbb{Z}}_{2^s}$-Linear Simplex and MacDonald Codes

The ${\mathbb{Z}}_{2^s}$-additive codes are subgroups of ${\mathbb{Z}}_{2^s}^n$, and can be seen as a generalization of linear codes over ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$. A ${\mathbb{Z}}_{2^s}$-linear code is a binary code which is the Gray map image of a ${\mathbb{Z}}_{2^s}$-additive code. We consider ${\mathbb{Z}}_{2^s}$-additive simplex codes of type $\alpha$ and $\beta$, which are a generalization over ${\mathbb{Z}}_{2^s}$ of the binary simplex codes. These ${\mathbb{Z}}_{2^s}$-additive simplex codes are related to the ${\mathbb{Z}}_{2^s}$-additive Hadamard codes. In this paper, we use this relationship to establish the kernel of their binary images, under the Gray map, the ${\mathbb{Z}}_{2^s}$-linear simplex codes. Similar results can be obtained for the binary Gray map image of ${\mathbb{Z}}_{2^s}$-additive MacDonald codes.