On One-Orbit Cyclic Subspace Codes of G q (n, 3)
Chiara Castello, Olga Polverino, Ferdinando Zullo · 2024
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$\text{Orb} (S)=\{\alpha S:\alpha\in \mathbb{F}_{q^{n}}^{\ast}\}$, 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 Orb(Fq3); 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.