On the full automorphism groups of $2$-designs constructed from finite fields ${\mathbb F}_{2^n}$

TL;DR

This paper investigates the automorphism groups of 2-designs derived from algebraic curves over finite fields, focusing on primitive maximal subgroups of the alternating group ${ m A}_{q-1}$ for fields ${ m GF}_{2^n}$.

Contribution

It identifies the full automorphism groups of these 2-designs as specific primitive maximal subgroups of the alternating group, providing new insights into their structure.

Findings

Automorphism groups are primitive maximal subgroups of ${ m A}_{q-1}$.

The 2-designs are constructed from algebraic curves over ${ m GF}_q$.

The study characterizes the automorphism groups explicitly.

Abstract

In this manuscript, for $q:=2^n$ with $n\geq2$, we study two primitive maximal subgroups of the alternating group ${\sf A}_{q-1}$. These subgroups are the full automorphism groups of $2$-designs which are constructed from algebraic curves over the finite field ${\mathbb F}_q$.