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arXiv · 2404.11241

Higher-dimensional grid-imprimitive block-transitive designs

Seyed Hassan Alavi·Carmen Amarra·Ashraf Daneshkhah·Alice Devillers·Cheryl E. Praeger
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Abstract

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.

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Seyed Hassan Alavi, Carmen Amarra, Ashraf Daneshkhah, Alice Devillers, Cheryl E. Praeger. 2024-04-17. Higher-dimensional grid-imprimitive block-transitive designs. https://arxiv.org/abs/2404.11241

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