New Infinite Families of $3$-Designs from Algebraic Curves of Higher Genus over Finite Fields

Abstract

In this paper, we give a simple method for computing the stabilizer subgroup of $D(f)=\{\alpha \in {\Bbb F}_q \mid \text{ there is a } \beta \in {\Bbb F}_q^{\times} \text{ such that }\beta^n=f(\alpha)\}$ in $PSL_2({\Bbb F}_q)$, where $q$ is a large odd prime power, $n$ is a positive integer dividing $q-1$ greater than $1$, and $f(x) \in {\Bbb F}_q[x]$. As an application, we construct new infinite families of $3$-designs.