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<p>It was shown in 1989 by Delandtsheer and Doyen that, for a 2-design with <i>v</i> points and block size <i>k</i>, a block-transitive group of automorphisms can be point-imprimitive (that is, leave invariant a nontrivial partition of the point set) only if <i>v</i> is small enough relative to <i>k</i>. 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.</p>

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Higher-dimensional grid-imprimitive block-transitive designs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月26日

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  • Seyed Hassan Alavi,
  • Carmen Amarra,
  • Ashraf Daneshkhah,
  • Alice Devillers,
  • Cheryl E. Praeger

摘要

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.