  We propose an innovative approach to investigating the linearity of
$\mathbb{Z}_{2^L}$-linear codes derived from $\mathbb{Z}_{2^L}$-additive codes
using the generalized Gray map. To achieve this, we define two related binary
codes: the associated and the decomposition codes. By considering the Schur
product between codewords, we can determine the linearity of the respective
$\mathbb{Z}_{2^L}$-linear code. As a result, we establish a connection between
the linearity of the $\mathbb{Z}_{2^L}$-linear codes with the linearity of the
decomposition code for $\mathbb{Z}_4$ and $\mathbb{Z}_8$-additive codes.
Furthermore, we construct $\mathbb{Z}_{2^L}$-additive codes from nested binary
codes, resulting in linear $\mathbb{Z}_{2^L}$-linear codes. This construction
involves multiple layers of binary codes, where a code in one layer is the
square of the code in the previous layer. We also employ our arguments to check
the linearity of well-known $\mathbb{Z}_{2^L}$-linear code constructions,
including the Hadamard, simplex, and MacDonald codes.
