Linearity of Z2L-linear codes via Schur product

We propose an innovative approach to investigating the linearity of Z2L$$\mathbb {Z}_{2^L}$$-linear codes derived from Z2L$$\mathbb {Z}_{2^L}$$-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 Z2L$$\mathbb {Z}_{2^L}$$-linear code. As a result, we establish a connection between the linearity of the Z2L$$\mathbb {Z}_{2^L}$$-linear codes with the linearity of the decomposition code for Z4$$\mathbb {Z}_4$$ and Z8$$\mathbb {Z}_8$$-additive codes. Furthermore, we construct Z2L$$\mathbb {Z}_{2^L}$$-additive codes from nested binary codes, resulting in linear Z2L$$\mathbb {Z}_{2^L}$$-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 Z2L$$\mathbb {Z}_{2^L}$$-linear codes by simple binary operations in their respective associated codes. Finally, we employ our arguments to verify the linearity of well-known Z2L$$\mathbb {Z}_{2^L}$$-linear code constructions, including the Hadamard, simplex, and MacDonald codes.