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 present a sufficient condition that allows checking nonlinearity of the \(\mathbb {Z}_{2^L}\) -linear codes by simple binary operations in their respective associated codes. Finally, we employ our arguments to verify the linearity of well-known \(\mathbb {Z}_{2^L}\) -linear code constructions, including the Hadamard, simplex, and MacDonald codes.
