Lifting of divisible designs

TL;DR

This paper introduces a new method called t-lifting to construct t-divisible designs for t>3 using finite projective spaces, filling a gap in the existing literature and providing numerous new examples.

Contribution

It develops an abstract construction method for t-divisible designs and provides explicit examples using projective spaces, cones, and Witt designs.

Findings

Constructed infinitely many non-isomorphic t-divisible designs for all t ≥ 2.

Developed a general t-lifting method starting from a t-divisible design and a group action.

Provided explicit examples using algebraic varieties and projective embeddings.

Abstract

The aim of this paper is to present a construction of $t$-divisible designs for $t>3$, because such divisible designs seem to be missing in the literature. To this end, tools such as finite projective spaces and their algebraic varieties are employed. More precisely, in a first step an abstract construction, called $t$-lifting, is developed. It starts from a set $X$ containing a $t$-divisible design and a group $G$ acting on $X$. Then several explicit examples are given, where $X$ is a subset of $PG(n,q)$ and $G$ is a subgroup of $GL_{n+1}(q)$. In some cases $X$ is obtained from a cone with a Veronesean or an $h$-sphere as its basis. In other examples $X$ arises from a projective embedding of a Witt design. As a result, for any integer $t\geq 2$ infinitely many non-isomorphic $t$-divisible designs are found.