Nonlinearity and Kernel of Z-Linear Simplex and MacDonald Codes

$\mathbb {Z}_{2^{s}}$
-additive codes are subgroups of
$\mathbb {Z}^{n}_{2^{s}}$
, 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 (not necessarily linear) 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 codes are related to the
$\mathbb {Z}_{2^{s}}$
-additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding
$\mathbb {Z}_{2^{s}}$
-linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly,
$\mathbb {Z}_{2^{s}}$
-additive MacDonald codes are defined for
$s>2$
, and equivalent results are obtained.