On point and block primitive designs invariant under permutation groups

TL;DR

This paper introduces a method for constructing and analyzing point and block primitive designs invariant under permutation groups, with implications for understanding the structure and enumeration of such designs.

Contribution

The paper presents a new construction method for point and block primitive G-invariant designs and discusses the theoretical possibility of their complete enumeration.

Findings

Method enables construction of primitive G-invariant designs

Every primitive G-invariant design can be generated by this method

Theoretically, all block transitive G-invariant designs can be identified

Abstract

In this paper, we present a method for constructing point primitive block transitive $t$-designs invariant under finite groups. Furthermore, we demonstrate that every point and block primitive $G$-invariant design can be generated using this method. Additionally, we establish the theoretical possibility of identifying all block transitive $G$-invariant designs. However, in practice, the feasibility of enumerating all designs for larger groups may be limited by the computational complexity involved.