We propose an original approach to investigate the linearity of Gray codes obtained from $\mathbb{Z}_{2^L}$-additive codes by introducing two related binary codes: the associated and concatenated. Once they are defined, one could perform a straightforward analysis of the Schur product between their codewords and determine the linearity of the respective Gray code. This work expands on earlier 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, MacDonald, Kerdock, and Preparata codes. We also present a family of Reed-Muller codes that yield to linear Gray 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，通过引入两种相关二进制码（即关联码与级联码），研究由$\mathbb{Z}_{2^L}$-加法码导出的格雷码的线性性质。一旦这些码被定义，即可直接分析其码字间的Schur乘积，并判定相应格雷码的线性性。本文拓展了文献中的早期贡献——此前线性性通常基于码的核或$\mathbb{Z}_{2^L}$上的运算建立。我们应用格雷映射并验证线性性的$\mathbb{Z}_{2^L}$-加法码包括著名的Hadamard码、单纯形码、MacDonald码、Kerdock码和Preparata码。此外，我们提出了一类能产生线性格雷码的Reed-Muller码族，并针对其他$\mathbb{Z}_{2^L}$-加法码对本文方法进行了计算验证。