Three-dimensional symmetric designs of propriety 3

TL;DR

This paper introduces a new class of three-dimensional symmetric designs with specific layer and intersection properties, providing necessary conditions, constructions, and classifications for these complex combinatorial objects.

Contribution

It generalizes higher-dimensional symmetric designs, establishes conditions for 3D designs, and constructs infinite families using various combinatorial structures.

Findings

Derived necessary conditions for parameters of 3D symmetric designs.

Constructed infinite families using difference sets, Hadamard matrices, and Latin cubes.

Enumerated small examples up to equivalence.

Abstract

We define symmetric designs of dimension $n$ and propriety $d$, providing a unifying generalization of several classes of higher-dimensional symmetric designs previously studied. We focus on the case $n=d=3$, which leads to the following question: Can we fill the $v^3$ cells of a $v\times v\times v$ cube with $\{0,1\}$ in such a way that each layer parallel to each face contains a fixed number $k$ of ones, and that for every two parallel layers there are exactly $\lambda$ positions where they have matching ones? We establish necessary conditions on the parameters $(v,k,\lambda)$, introduce notions of difference sets and multipliers for these objects, and enumerate small examples up to equivalence. Furthermore, we construct infinite families of these objects using difference sets, symmetric designs, doubly regular tournaments, Hadamard matrices, Latin cubes, and association schemes on triples.