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 employ our arguments to check the linearity of well-known \(\mathbb{Z}_{2^L}\)-linear code constructions, including the Hadamard, simplex, and MacDonald codes.