On $3$-designs from $PGL(2,q)$

TL;DR

This paper investigates the construction of 3-designs from the action of the projective linear group PGL(2,q) on the projective line, identifying parameters for specific orbit-based designs.

Contribution

It determines the parameters of 3-designs formed by orbits of certain blocks under PGL(2,q) actions, expanding understanding of combinatorial designs from group actions.

Findings

Parameters of 3-designs from PGL(2,q) orbits are explicitly derived.

Identifies specific block structures leading to 3-designs.

Provides new examples of combinatorial designs from group actions.

Abstract

The group $PGL(2,q)$ acts $3$-transitively on the projective line $GF(q) \cup \{\infty\}$. Thus, an orbit of its action on the $k$-subsets of the projective line is the block set of a $3$-$(q+1,k,\lambda)$ design. We find the parameters of the designs formed by the orbit of a block of the form $\langle \theta^r \rangle$ or $\langle \theta^r \rangle \cup \{ 0\}$, where $\theta$ is a primitive element of $GF(q)$.