Linearized Reed-Solomon (LRS) codes form an important family of maximum sum-rank distance (MSRD) codes that generalize both Reed--Solomon codes and Gabidulin codes. In this paper we study the equivalence problem for LRS codes and determine the number of inequivalent codes within this family. Using the correspondence between sum-rank metric codes and systems of $\mathbb{F}_q$-subspaces, we analyze the stabilizer of the Gabidulin system and derive a characterization of equivalence between LRS codes. In particular, we prove that two LRS codes are equivalent if and only if the sets of norms that define the codes coincide up to multiplication by an element of $\mathbb{F}_q^\ast$. This description allows us to reduce the classification problem to the action of $\mathbb{F}_q^\ast$ on subsets of $\mathbb{F}_q^\ast$. As a consequence, we derive formulas for the number of inequivalent linearized Reed-Solomon codes and illustrate the results with explicit examples.

翻译：线性化Reed-Solomon（LRS）码是最大和秩距离（MSRD）码的重要族系，它同时推广了Reed-Solomon码与Gabidulin码。本文研究LRS码的等价问题，并确定该族系内非等价码的数量。利用和秩度量码与$\mathbb{F}_q$-子空间系统之间的对应关系，我们分析了Gabidulin系统的稳定化子，推导出LRS码等价的刻画条件。具体而言，我们证明两个LRS码等价当且仅当定义这些码的范数集在$\mathbb{F}_q^\ast$元素乘法下一致。该描述将分类问题简化为$\mathbb{F}_q^\ast$作用于$\mathbb{F}_q^\ast$子集上的作用。由此，我们推导出非等价线性化Reed-Solomon码数量的公式，并通过具体实例阐释结果。