We propose an original approach to investigate the linearity of $\mathbb{Z}_{2^L}$-linear codes, i.e., codes obtained as the image of the generalized Gray map applied to $\mathbb{Z}_{2^L}$-additive codes. To accomplish that, we define two related binary codes: the associated and concatenated, where one could perform a straightforward analysis of the Schur product between their codewords and determine the linearity of the respective $\mathbb{Z}_{2^L}$-linear code. This work expands on previous contributions from the literature, where the linearity was established with respect to the kernel of a code and/or operations on $\mathbb{Z}_{2^L}$. The $\mathbb{Z}_{2^L}$-additive codes we apply the Gray map and check the linearity are the well-known Hadamard, simplex, and MacDonald codes. We also present families of Reed-Muller and cyclic codes that yield to linear $\mathbb{Z}_{2^L}$-linear codes and perform a computational verification of our proposed method applied to other $\mathbb{Z}_{2^L}$-additive codes.

翻译：我们提出了一种原创方法，用于研究$\mathbb{Z}_{2^L}$-线性码（即通过对$\mathbb{Z}_{2^L}$-加法码应用广义Gray映射所得到的码）的线性性。为此，我们定义了两种相关的二进制码：关联码与级联码。利用这两种码，我们可以直接分析其码字之间的Schur积，从而确定相应$\mathbb{Z}_{2^L}$-线性码的线性性。本研究在已有文献成果基础上进行了扩展——此前文献中线性性的建立依赖于码的核结构及/或在$\mathbb{Z}_{2^L}$上的运算。我们应用Gray映射并检验线性性的$\mathbb{Z}_{2^L}$-加法码包括著名的Hadamard码、Simplex码和MacDonald码。此外，我们给出了若干能够生成线性$\mathbb{Z}_{2^L}$-线性码的Reed-Muller码和循环码族，并通过计算验证了所提方法在其他$\mathbb{Z}_{2^L}$-加法码上的应用效果。