On one-orbit cyclic subspace codes of ${\mathcal{G}}_q(n,3)$

Abstract

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 \[ \mathrm{Orb}(S)=\{αS \colon α\in \mathbb{F}_{q^n}^*\}, \]where $\alpha S=\{\alpha s:s\in S\}$ for any $\alpha \in {\mathbb{F}}_{q^n}^{\ast }$. 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.

Accepted to the 2024 IEEE International Symposium on Information Theory (ISIT 2024)