Simple $3$-Designs of PS $L(2,2^n)$ with Block Size $7$

Abstract

This paper devotes to the investigation of $3$-designs admitting the special projective linear group $\mathrm{P}\mathrm{S}\mathrm{L}(2,2^n)$ as an automorphism, and we determine all the possible values of $n$ in the simple $3-(2^n+1,7,\lambda )$ designs admitting $\mathrm{P}\mathrm{S}\mathrm{L}(2,2^n)$ as an automorphism group.