Partial Geometric Designs from Group Actions

TL;DR

This paper introduces a novel method using group actions to construct infinite families of partial geometric designs and balanced incomplete block designs, expanding the toolkit for combinatorial design construction.

Contribution

It presents a new approach leveraging group actions to generate partial geometric designs and BIBDs, with explicit constructions and analysis of stabilizers in linear groups.

Findings

Constructed several infinite families of partial geometric designs.

Developed a new infinite family of balanced incomplete block designs.

Analyzed stabilizers of subsets in linear groups of degree two.

Abstract

In this paper, using group actions, we introduce a new method for constructing partial geometric designs (sometimes referred to as $1\frac{1}{2}$-designs). Using this new method, we construct several infinite families of partial geometric designs by investigating the actions of various linear groups of degree two on certain subsets of $\mathbb{F}_{q}^{2}$. Moreover, by computing the stabilizers of such subsets in various linear groups of degree two, we are also able to construct a new infinite family of balanced incomplete block designs.