Projection cubes of symmetric designs

TL;DR

This paper introduces a new n-dimensional generalization of symmetric block designs, establishing bounds, defining difference sets, extending classic families, and classifying small cases.

Contribution

It presents the concept of n-dimensional symmetric designs and difference sets, extending classical combinatorial structures to higher dimensions with new bounds and classifications.

Findings

Established upper bounds on the dimension n

Extended classic difference sets to higher dimensions

Classified small parameter cases

Abstract

We introduce a new type of $n$-dimensional generalization of symmetric $(v,k,\lambda)$ 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 $(v,k,\lambda)$ and some interesting examples are presented.