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arXiv · 2412.09067

On higher-dimensional symmetric designs

Vedran Krčadinac·Mario Osvin Pavčević
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Abstract

We study two kinds of generalizations of symmetric block designs to higher dimensions, the so-called $\mathcal{C}$-cubes and $\mathcal{P}$-cubes. For small parameters, all examples up to equivalence are determined by computer calculations. Known properties of automorphisms of symmetric designs are extended to autotopies of $\mathcal{P}$-cubes, while counterexamples are found for $\mathcal{C}$-cubes. An algorithm for the classification of $\mathcal{P}$-cubes with prescribed autotopy groups is developed and used to construct more examples. A bound on the dimension of difference sets for $\mathcal{P}$-cubes is proved and shown to be tight in elementary abelian groups. The construction is generalized to arbitrary groups by introducing regular sets of (anti)automorphisms.

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Vedran Krčadinac, Mario Osvin Pavčević. 2024-12-12. On higher-dimensional symmetric designs. https://doi.org/10.1080/10586458.2025.2521450

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