Three-dimensional symmetric designs of propriety 3

TL;DR

This work tackles the problem of three-dimensional symmetric designs of propriety $3$, formalizing the object as $\mathcal{C}^3_3(v,k,\lambda)$ and examining the parameter constraints, constructions, and interconnections with classical combinatorial objects. The authors develop a comprehensive framework linking to incidence designs, difference sets, Hadamard matrices, Latin cubes, and association schemes on triples, and they derive key necessary inequalities such as $\lambda v(v-1) \ge k(k-v)$ with equality characterizing special cases. They present infinite families of $\mathcal{C}^3_3(v,k,\lambda)$-cubes via product constructions, layer-rainbow Hadamard-based methods, and AST-based schemes, and they provide explicit parameter formulas and numerous examples, including many nonequivalent cubes for fixed $(v,k,\lambda)$. Their work broadens the landscape of higher-dimensional symmetric designs, linking to diverse combinatorial structures and offering both theoretical insight and computational data to guide future exploration and potential applications in design theory and related fields.

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 $λ$ positions where they have matching ones? We establish necessary conditions on the parameters $(v,k,λ)$, 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.

Figures (5)

• Figure 1: The identity cube $I\in \mathcal{C}^3_3(4,1,0)$.
• Figure 2: Two nonequivalent $\mathcal{C}^3_3(5,5,0)$-cubes and the corresponding Latin squares.
• Figure 3: Two cubes in $\mathcal{C}^3_3(7,3,0)$.
• Figure 4: Two cubes in $\mathcal{C}^3_3(7,21,7)$.
• Figure 5: The cube of Corollary \ref{['trivrel']} for $v=7$.

Theorems & Definitions (66)

• Definition 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof