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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.02200 |
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| _version_ | 1866914855818100736 |
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| author | Mahak Bhaintwal, Maheshanand |
| author_facet | Mahak Bhaintwal, Maheshanand |
| contents | {A cyclic subspace code is a union of the orbits of subspaces contained in it. In a recent paper, Gluesing-Luerssen et al. (Des. Codes Cryptogr. 89, 447-470, 2021) showed that the study of the distance distribution of a single orbit cyclic subspace code is equivalent to the study of its intersection distribution. In this paper we have proved that in the orbit of a subspace $U$ of $\mathbb{F}_{q^n}$ that has the stabilizer $\mathbb{F}_{q^t}^*(t \neq n)$, the number of codeword pairs $(U,αU)$ such that $\dim(U\cap αU)=i$ for any $i,~ 0\leq i < \dim(U)$, is a multiple of $q^t(q^t+1)$, if $\frac{n}{t}$ is an odd number. In the case of even $\frac{n}{t}$, if $U$ contains $\frac{q^{2tm}-1}{q^{2t}-1}~ (m\geq 0)$ distinct cyclic shifts of $\mathbb{F}_{q^{2t}}$, then the number of codeword pairs $(U,αU)$ with intersection dimension $2tm$ is equal to $q^t+rq^t(q^t+1)$, for some non-negative integer $r$; and the number of codeword pairs $(U,αU)$ with intersection dimension $i,~(i\neq 2tm)$ is a multiple of $q^t(q^t+1)$. Some examples have been given to illustrate the results presented in the paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02200 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the distance distributions of single-orbit cyclic subspace codes Mahak Bhaintwal, Maheshanand Information Theory {A cyclic subspace code is a union of the orbits of subspaces contained in it. In a recent paper, Gluesing-Luerssen et al. (Des. Codes Cryptogr. 89, 447-470, 2021) showed that the study of the distance distribution of a single orbit cyclic subspace code is equivalent to the study of its intersection distribution. In this paper we have proved that in the orbit of a subspace $U$ of $\mathbb{F}_{q^n}$ that has the stabilizer $\mathbb{F}_{q^t}^*(t \neq n)$, the number of codeword pairs $(U,αU)$ such that $\dim(U\cap αU)=i$ for any $i,~ 0\leq i < \dim(U)$, is a multiple of $q^t(q^t+1)$, if $\frac{n}{t}$ is an odd number. In the case of even $\frac{n}{t}$, if $U$ contains $\frac{q^{2tm}-1}{q^{2t}-1}~ (m\geq 0)$ distinct cyclic shifts of $\mathbb{F}_{q^{2t}}$, then the number of codeword pairs $(U,αU)$ with intersection dimension $2tm$ is equal to $q^t+rq^t(q^t+1)$, for some non-negative integer $r$; and the number of codeword pairs $(U,αU)$ with intersection dimension $i,~(i\neq 2tm)$ is a multiple of $q^t(q^t+1)$. Some examples have been given to illustrate the results presented in the paper. |
| title | On the distance distributions of single-orbit cyclic subspace codes |
| topic | Information Theory |
| url | https://arxiv.org/abs/2407.02200 |