On one-orbit cyclic subspace codes of $\mathcal{G}_q(n,3)$
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arXiv
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| Format: | Preprint |
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2024
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| author | Castello, Chiara Polverino, Olga Zullo, Ferdinando |
| author_facet | Castello, Chiara Polverino, Olga Zullo, Ferdinando |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On one-orbit cyclic subspace codes of $\mathcal{G}_q(n,3)$ Castello, Chiara Polverino, Olga Zullo, Ferdinando Information Theory Combinatorics 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. |
| title | On one-orbit cyclic subspace codes of $\mathcal{G}_q(n,3)$ |
| topic | Information Theory Combinatorics |
| url | https://arxiv.org/abs/2405.01652 |