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

TL;DR

This paper classifies three-dimensional one-orbit cyclic subspace codes over finite fields, focusing on their structure, equivalence, and properties related to minimum distance, contributing to the understanding of their role in network coding.

Contribution

It provides a classification of three-dimensional one-orbit cyclic subspace codes, analyzing their families and inequivalence, advancing the understanding of their structure and applications.

Findings

Identified three families of three-dimensional one-orbit cyclic subspace codes.

Analyzed inequivalence within the families with optimum and minimum distance.

Contributed to the classification of cyclic subspace codes in the context of network coding.

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 \[ \mathrm{Orb}(S)=\{\alpha S \colon \alpha \in \mathbb{F}_{q^n}^*\}, \]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.