Linearity of $\mathbb{Z}_{2^L}$-Linear Codes via Schur Product

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.