On higher-dimensional symmetric designs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911180137693184 |
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| author | Krčadinac, Vedran Pavčević, Mario Osvin |
| author_facet | Krčadinac, Vedran Pavčević, Mario Osvin |
| contents | We study two kinds of generalizations of symmetric block designs to higher dimensions, the so-called $\mathcal{C}$-cubes and $\mathcal{P}$-cubes. For small parameters, all examples up to equivalence are determined by computer calculations. Known properties of automorphisms of symmetric designs are extended to autotopies of $\mathcal{P}$-cubes, while counterexamples are found for $\mathcal{C}$-cubes. An algorithm for the classification of $\mathcal{P}$-cubes with prescribed autotopy groups is developed and used to construct more examples. A bound on the dimension of difference sets for $\mathcal{P}$-cubes is proved and shown to be tight in elementary abelian groups. The construction is generalized to arbitrary groups by introducing regular sets of (anti)automorphisms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09067 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On higher-dimensional symmetric designs Krčadinac, Vedran Pavčević, Mario Osvin Combinatorics 05B05, 05B10, 05B20 We study two kinds of generalizations of symmetric block designs to higher dimensions, the so-called $\mathcal{C}$-cubes and $\mathcal{P}$-cubes. For small parameters, all examples up to equivalence are determined by computer calculations. Known properties of automorphisms of symmetric designs are extended to autotopies of $\mathcal{P}$-cubes, while counterexamples are found for $\mathcal{C}$-cubes. An algorithm for the classification of $\mathcal{P}$-cubes with prescribed autotopy groups is developed and used to construct more examples. A bound on the dimension of difference sets for $\mathcal{P}$-cubes is proved and shown to be tight in elementary abelian groups. The construction is generalized to arbitrary groups by introducing regular sets of (anti)automorphisms. |
| title | On higher-dimensional symmetric designs |
| topic | Combinatorics 05B05, 05B10, 05B20 |
| url | https://arxiv.org/abs/2412.09067 |