A new series of large sets of subspace designs over the binary field
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2016
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| _version_ | 1866916980520386560 |
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| author | Kiermaier, Michael Laue, Reinhard Wassermann, Alfred |
| author_facet | Kiermaier, Michael Laue, Reinhard Wassermann, Alfred |
| contents | In this article, we show the existence of large sets $\operatorname{LS}_2[3](2,k,v)$ for infinitely many values of $k$ and $v$. The exact condition is $v \geq 8$ and $0 \leq k \leq v$ such that for the remainders $\bar{v}$ and $\bar{k}$ of $v$ and $k$ modulo $6$ we have $2 \leq \bar{v} < \bar{k} \leq 5$.
The proof is constructive and consists of two parts. First, we give a computer construction for an $\operatorname{LS}_2[3](2,4,8)$, which is a partition of the set of all $4$-dimensional subspaces of an $8$-dimensional vector space over the binary field into three disjoint $2$-$(8, 4, 217)_2$ subspace designs. Together with the already known $\operatorname{LS}_2[3](2,3,8)$, the application of a recursion method based on a decomposition of the Graßmannian into joins yields a construction for the claimed large sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1603_06976 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | A new series of large sets of subspace designs over the binary field Kiermaier, Michael Laue, Reinhard Wassermann, Alfred Combinatorics In this article, we show the existence of large sets $\operatorname{LS}_2[3](2,k,v)$ for infinitely many values of $k$ and $v$. The exact condition is $v \geq 8$ and $0 \leq k \leq v$ such that for the remainders $\bar{v}$ and $\bar{k}$ of $v$ and $k$ modulo $6$ we have $2 \leq \bar{v} < \bar{k} \leq 5$. The proof is constructive and consists of two parts. First, we give a computer construction for an $\operatorname{LS}_2[3](2,4,8)$, which is a partition of the set of all $4$-dimensional subspaces of an $8$-dimensional vector space over the binary field into three disjoint $2$-$(8, 4, 217)_2$ subspace designs. Together with the already known $\operatorname{LS}_2[3](2,3,8)$, the application of a recursion method based on a decomposition of the Graßmannian into joins yields a construction for the claimed large sets. |
| title | A new series of large sets of subspace designs over the binary field |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1603.06976 |