On One-Orbit Cyclic Subspace Codes of Gq(n, 3)

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(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.