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 Fq-subspace of Fqⁿ, then the one-orbit cyclic subspace code defined by S is \[ Orb(S)=\{α S α ∈ Fq^n^*\}, \]where α S= α s s∈ S for any α∈ Fqⁿ^*. 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.