Simple $3$-designs of $\mathrm{P}\mathrm{S}\mathrm{L}(2,2^n)$ with block size $13$

This paper investigates simple $3$-$(2^n+1,13,\lambda )$ designs admitting $\mathrm{P}\mathrm{S}\mathrm{L}$$(2,2^n)$ as an automorphism group. We determine all possible values of $\lambda$ by systematically analyzing the orbits of $13$-element subsets under the action of $\mathrm{P}\mathrm{S}\mathrm{L}$$(2,2^n)$ on the projective line. While previous research has explored this topic by analyzing the structure of $k$-element subsets $B$ directly, we approach the problem using group theory and the Cauchy-Frobenius-Burnside lemma. This method provides an efficient framework that can be applied to larger block sizes where traditional enumeration methods become computationally infeasible.