Lifting of divisible designs

TL;DR

The aim of this paper is to present a construction of t-divisible designs (DDs) for t > 3, because such DDs seem to be missing in the literature, and tools such as finite projective spaces and their algebraic varieties are employed.

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.

Theorems & Definitions (13)

• Definition 2.2
• Theorem 2.5: $t$-Lifting
• Theorem 2.6
• Lemma 2.7
• Theorem 2.8
• Theorem 3.1
• Corollary 3.3
• Theorem 3.4
• Lemma 3.7
• Theorem 3.8