On higher-dimensional symmetric designs

TL;DR

This paper explores higher-dimensional generalizations of symmetric block designs, introduces new concepts like $\\mathcal{C}$-cubes and $\\\mathcal{P}$-cubes, and develops algorithms for their classification and analysis.

Contribution

It extends properties of symmetric designs to higher dimensions, develops classification algorithms for $\\\mathcal{P}$-cubes, and establishes bounds on difference sets.

Findings

All small examples classified by computer

Automorphism properties extended to autotopies of $\\\mathcal{P}$-cubes

Bound on difference set dimension proved and shown to be tight

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.