Higher-dimensional grid-imprimitive block-transitive designs

• Published in: arXiv.org 2024-04
• Main Authors: Seyed Hassan Alavi, Amarra, Carmen, Daneshkhah, Ashraf, Devillers, Alice, Praeger, Cheryl E
• Format: Article
• Language: English
• Subjects: Automorphisms; Parameters
• ISSN: 2331-8422

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.