Subspace codes have recently been used for error correction in random network coding. In this work, we focus on one-orbit cyclic subspace codes. If $S$ is an $\mathbb{F}_q$-subspace of $\mathbb{F}_{q^n}$, then the one-orbit cyclic subspace code defined by $S$ is \[ \mathrm{Orb}(S)=\{\alpha S \colon \alpha \in \mathbb{F}_{q^n}^*\}, \]where $\alpha S=\lbrace \alpha s \colon s\in S\rbrace$ for any $\alpha\in \mathbb{F}_{q^n}^*$. Few classification results of subspace codes are known, therefore it is quite natural to initiate a classification of cyclic subspace codes, especially in the light of the recent classification of the isometries for cyclic subspace codes. We consider three-dimensional one-orbit cyclic subspace codes, which are divided into three families: the first one containing only $\mathrm{Orb}(\mathbb{F}_{q^3})$; the second one containing the optimum-distance codes; and the third one whose elements are codes with minimum distance $2$. We study inequivalent codes in the latter two families.

翻译：子空间码最近被用于随机网络编码中的纠错。本研究聚焦于单轨道循环子空间码。若 $S$ 是 $\mathbb{F}_{q^n}$ 的 $\mathbb{F}_q$-子空间，则由 $S$ 定义的单轨道循环子空间码为 \[ \mathrm{Orb}(S)=\{\alpha S \colon \alpha \in \mathbb{F}_{q^n}^*\}，\]其中对任意 $\alpha\in \mathbb{F}_{q^n}^*$，$\alpha S=\lbrace \alpha s \colon s\in S\rbrace$。目前子空间码的分类结果较少，因此对循环子空间码进行分类具有天然合理性，尤其是在近期循环子空间码等距分类工作的推动下。我们考虑三维单轨道循环子空间码，将其分为三类：第一类仅包含 $\mathrm{Orb}(\mathbb{F}_{q^3})$；第二类包含最优距离码；第三类元素为最小距离为 $2$ 的码。本研究探讨后两类中的非等价码。