Large sets of subspace designs
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2014
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| _version_ | 1866909816953241600 |
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| author | Braun, Michael Kiermaier, Michael Kohnert, Axel Laue, Reinhard |
| author_facet | Braun, Michael Kiermaier, Michael Kohnert, Axel Laue, Reinhard |
| contents | In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Graßmannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs.
We construct a $2$-$(6,3,78)_5$ design by computer, which corresponds to a halving $\operatorname{LS}_5[2](2,3,6)$. The application of the new recursion method to this halving and an already known $\operatorname{LS}_3[2](2,3,6)$ yields two infinite two-parameter series of halvings $\operatorname{LS}_3[2](2,k,v)$ and $\operatorname{LS}_5[2](2,k,v)$ with integers $v\geq 6$, $v\equiv 2\mod 4$ and $3\leq k\leq v-3$, $k\equiv 3\mod 4$.
Thus in particular, two new infinite series of nontrivial subspace designs with $t = 2$ are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with $t = 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1411_7181 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Large sets of subspace designs Braun, Michael Kiermaier, Michael Kohnert, Axel Laue, Reinhard Combinatorics Primary 51E20, Secondary 05B05, 05B25, 11Txx In this article, three types of joins are introduced for subspaces of a vector space. Decompositions of the Graßmannian into joins are discussed. This framework admits a generalization of large set recursion methods for block designs to subspace designs. We construct a $2$-$(6,3,78)_5$ design by computer, which corresponds to a halving $\operatorname{LS}_5[2](2,3,6)$. The application of the new recursion method to this halving and an already known $\operatorname{LS}_3[2](2,3,6)$ yields two infinite two-parameter series of halvings $\operatorname{LS}_3[2](2,k,v)$ and $\operatorname{LS}_5[2](2,k,v)$ with integers $v\geq 6$, $v\equiv 2\mod 4$ and $3\leq k\leq v-3$, $k\equiv 3\mod 4$. Thus in particular, two new infinite series of nontrivial subspace designs with $t = 2$ are constructed. Furthermore as a corollary, we get the existence of infinitely many nontrivial large sets of subspace designs with $t = 2$. |
| title | Large sets of subspace designs |
| topic | Combinatorics Primary 51E20, Secondary 05B05, 05B25, 11Txx |
| url | https://arxiv.org/abs/1411.7181 |