On one-orbit cyclic subspace codes of $\mathcal{G}_q(n,3)$
Chiara Castello, Olga Polverino, Ferdinando Zullo
TL;DR
The paper investigates 3-dimensional one-orbit cyclic subspace codes in the Grassmannian ${\mathcal G}_q(n,3)$, aiming to classify inequivalent codes under semilinear equivalence. It develops invariants based on square-span and larger-field spans to distinguish codes, and provides a detailed structural classification of subspaces $S$ by the dimension of $S^2$ and the resulting code distance, including Sidon-space cases. For optimum-distance codes with distance $4$, the authors derive a linearized-polynomial description $S=V_{f,\gamma}$ and reduce equivalence to the ${\mathcal V}_{U,\gamma}$ framework, showing that appropriate representatives are $V_{x^q,\gamma}$ or $V_{\mathrm{Tr}_{{\mathbb F}_{q^3}/{\mathbb F_q}}(x),\gamma}$ (with additional conditions when $n=6$). The results advance the classification of cyclic subspace codes and illuminate constructions with strong error-correction properties for random 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 \[ \mathrm{Orb}(S)=\{αS \colon α\in \mathbb{F}_{q^n}^*\}, \]where $αS=\lbrace αs \colon s\in S\rbrace$ for any $α\in \mathbb{F}_{q^n}^*$. 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.