🤖 AI Summary
This work addresses the linearity testing problem for $mathbb{Z}_{2^L}$-linear codes: given a code derived from a $mathbb{Z}_{2^L}$-additive code via the generalized Gray map, how to efficiently determine whether it is $mathbb{Z}_{2^L}$-linear. To this end, we introduce the notions of *associated codes* and *decomposed codes*, and establish—first time—the equivalence between $mathbb{Z}_4$- and $mathbb{Z}_8$-linearity and the linearity of their decomposed counterparts. For general $L$, we derive an algebraic criterion using the Schur product. Furthermore, we propose a novel nested binary construction (via iterative squaring) that systematically generates diverse new families of linear $mathbb{Z}_{2^L}$-codes. Finally, we completely resolve the linearity characterization for classical families—including Hadamard, Simplex, and MacDonald codes—thereby unifying and extending the theoretical framework and constructive methodology for $mathbb{Z}_{2^L}$-linear codes.
📝 Abstract
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.