Projection cubes of symmetric designs

TL;DR

This work introduces projection $n$-cubes, a higher-dimensional generalization of symmetric $(v,k,\lambda)$ designs defined by the requirement that every $2$-dimensional projection is a symmetric design. It establishes key connections to orthogonal arrays $OA(vk,n,v,1)$ and derives dimension bounds, showing $n$ is finite and bounded by expressions in $v$ and $k$. The authors develop $n$-dimensional difference sets and extend classic finite-field constructions—Paley, cyclotomic, and twin prime power differences—to yield large families of projection cubes, while also performing exhaustive classifications for small parameters and identifying new non-difference-cube examples. Computational and theoretical results together reveal a rich structure: some projection cubes arise from difference sets with sharply transitive autotopy groups, while others do not, especially in non-abelian settings. The work lays groundwork for broader classification, construction of infinite families, and exploration of projection cubes beyond difference-set origins, with practical implications for combinatorial design theory and OA-based representations.

Abstract

We introduce a new type of $n$-dimensional generalization of symmetric $(v,k,λ)$ block designs. We prove upper bounds on the dimension $n$ in terms of $v$ and $k$. We also define the corresponding concept of $n$-dimensional difference sets, and extend some classic families of difference sets to higher dimensions. Complete classifications are performed for small parameters $(v,k,λ)$ and some interesting examples are presented.

Figures (2)

• Figure 1: Two disjoint cubes in $\mathcal{P}^3(7,3,1)$.
• Figure 2: Cubes in $\mathcal{P}^3(16,6,2)$ with non-isomorphic projections.

Theorems & Definitions (27)

• Definition 2.1
• Example 2.2
• Proposition 2.3
• Proposition 2.4
• proof
• Corollary 2.5
• Example 2.6
• Example 2.7
• Theorem 2.8
• proof