Three-dimensional symmetric designs of propriety 3
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908602966474752 |
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| author | Bahmanian, Amin Krčadinac, Vedran Relić, Lucija Suda, Sho |
| author_facet | Bahmanian, Amin Krčadinac, Vedran Relić, Lucija Suda, Sho |
| contents | We define symmetric designs of dimension $n$ and propriety $d$, providing a unifying generalization of several classes of higher-dimensional symmetric designs previously studied. We focus on the case $n=d=3$, which leads to the following question: Can we fill the $v^3$ cells of a $v\times v\times v$ cube with $\{0,1\}$ in such a way that each layer parallel to each face contains a fixed number $k$ of ones, and that for every two parallel layers there are exactly $λ$ positions where they have matching ones? We establish necessary conditions on the parameters $(v,k,λ)$, introduce notions of difference sets and multipliers for these objects, and enumerate small examples up to equivalence. Furthermore, we construct infinite families of these objects using difference sets, symmetric designs, doubly regular tournaments, Hadamard matrices, Latin cubes, and association schemes on triples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17337 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Three-dimensional symmetric designs of propriety 3 Bahmanian, Amin Krčadinac, Vedran Relić, Lucija Suda, Sho Combinatorics 05B05, 05E30, 05B20, 05B15 We define symmetric designs of dimension $n$ and propriety $d$, providing a unifying generalization of several classes of higher-dimensional symmetric designs previously studied. We focus on the case $n=d=3$, which leads to the following question: Can we fill the $v^3$ cells of a $v\times v\times v$ cube with $\{0,1\}$ in such a way that each layer parallel to each face contains a fixed number $k$ of ones, and that for every two parallel layers there are exactly $λ$ positions where they have matching ones? We establish necessary conditions on the parameters $(v,k,λ)$, introduce notions of difference sets and multipliers for these objects, and enumerate small examples up to equivalence. Furthermore, we construct infinite families of these objects using difference sets, symmetric designs, doubly regular tournaments, Hadamard matrices, Latin cubes, and association schemes on triples. |
| title | Three-dimensional symmetric designs of propriety 3 |
| topic | Combinatorics 05B05, 05E30, 05B20, 05B15 |
| url | https://arxiv.org/abs/2510.17337 |