Higher-dimensional grid-imprimitive block-transitive designs
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917642450763776 |
|---|---|
| author | Alavi, Seyed Hassan Amarra, Carmen Daneshkhah, Ashraf Devillers, Alice Praeger, Cheryl E. |
| author_facet | Alavi, Seyed Hassan Amarra, Carmen Daneshkhah, Ashraf Devillers, Alice Praeger, Cheryl E. |
| contents | It was shown in 1989 by Delandtsheer and Doyen that, for a $2$-design with $v$ points and block size $k$, a block-transitive group of automorphisms can be point-imprimitive (that is, leave invariant a nontrivial partition of the point set) only if $v$ is small enough relative to $k$. Recently, exploiting a construction of block-transitive point-imprimitive $2$-designs given by Cameron and the last author, four of the authors studied $2$-designs admitting a block-transitive group that preserves a two-dimensional grid structure on the point set. Here we consider the case where there a block-transitive group preserves a multidimensional grid structure on points. We provide necessary and sufficient conditions for such $2$-designs to exist in terms of the parameters of the grid, and certain `array parameters' which describe a subset of points (which will be a block of the design). Using this criterion, we construct explicit examples of $2$-designs for grids of dimensions three and four, and pose several open questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher-dimensional grid-imprimitive block-transitive designs Alavi, Seyed Hassan Amarra, Carmen Daneshkhah, Ashraf Devillers, Alice Praeger, Cheryl E. Combinatorics 05B05, 05B25, 20B25 It was shown in 1989 by Delandtsheer and Doyen that, for a $2$-design with $v$ points and block size $k$, a block-transitive group of automorphisms can be point-imprimitive (that is, leave invariant a nontrivial partition of the point set) only if $v$ is small enough relative to $k$. Recently, exploiting a construction of block-transitive point-imprimitive $2$-designs given by Cameron and the last author, four of the authors studied $2$-designs admitting a block-transitive group that preserves a two-dimensional grid structure on the point set. Here we consider the case where there a block-transitive group preserves a multidimensional grid structure on points. We provide necessary and sufficient conditions for such $2$-designs to exist in terms of the parameters of the grid, and certain `array parameters' which describe a subset of points (which will be a block of the design). Using this criterion, we construct explicit examples of $2$-designs for grids of dimensions three and four, and pose several open questions. |
| title | Higher-dimensional grid-imprimitive block-transitive designs |
| topic | Combinatorics 05B05, 05B25, 20B25 |
| url | https://arxiv.org/abs/2404.11241 |