Lifting of divisible designs
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2013
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| _version_ | 1866909090347745280 |
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| author | Blunck, Andrea Havlicek, Hans Zanella, Corrado |
| author_facet | Blunck, Andrea Havlicek, Hans Zanella, Corrado |
| contents | The aim of this paper is to present a construction of $t$-divisible designs for $t>3$, because such divisible designs seem to be missing in the literature. To this end, tools such as finite projective spaces and their algebraic varieties are employed. More precisely, in a first step an abstract construction, called $t$-lifting, is developed. It starts from a set $X$ containing a $t$-divisible design and a group $G$ acting on $X$. Then several explicit examples are given, where $X$ is a subset of $PG(n,q)$ and $G$ is a subgroup of $GL_{n+1}(q)$. In some cases $X$ is obtained from a cone with a Veronesean or an $h$-sphere as its basis. In other examples $X$ arises from a projective embedding of a Witt design. As a result, for any integer $t\geq 2$ infinitely many non-isomorphic $t$-divisible designs are found. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1304_1337 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Lifting of divisible designs Blunck, Andrea Havlicek, Hans Zanella, Corrado Combinatorics Rings and Algebras 05B30, 51E20, 20B25 The aim of this paper is to present a construction of $t$-divisible designs for $t>3$, because such divisible designs seem to be missing in the literature. To this end, tools such as finite projective spaces and their algebraic varieties are employed. More precisely, in a first step an abstract construction, called $t$-lifting, is developed. It starts from a set $X$ containing a $t$-divisible design and a group $G$ acting on $X$. Then several explicit examples are given, where $X$ is a subset of $PG(n,q)$ and $G$ is a subgroup of $GL_{n+1}(q)$. In some cases $X$ is obtained from a cone with a Veronesean or an $h$-sphere as its basis. In other examples $X$ arises from a projective embedding of a Witt design. As a result, for any integer $t\geq 2$ infinitely many non-isomorphic $t$-divisible designs are found. |
| title | Lifting of divisible designs |
| topic | Combinatorics Rings and Algebras 05B30, 51E20, 20B25 |
| url | https://arxiv.org/abs/1304.1337 |