On 3-Designs From P G L ( 2 , q )

Abstract

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