Simple $3$-designs of $\mathrm{PSL}(2,2^n)$ with block size $13$
Takara Kondo, Yuto Nogata
TL;DR
This work classifies simple $3$-designs with block size $k=13$ that admit the automorphism group $G=\mathrm{PSL}(2,2^n)$ acting on the projective line $X=\mathbb{F}_{2^n}\cup\{\infty\}$. By exploiting the sharply $3$-transitive action of $G$, the authors reduce the problem to orbit decompositions of $13$-subsets and apply the Cauchy-Frobenius-Burnside lemma to count orbits and stabilize designs without enumerating blocks. They identify feasible stabilizer orders $|G_B|\in\{1,2,3,4,12,22,26\}$, derive the corresponding $\lambda_B$ values, and compute the multiplicities $m_{\lambda_B}$ for $\lambda_B\in\{66,78,143,429,572,858,1716\}$, with results depending on $n$ modulo $60$. The main theorem then expresses the existence of a simple $3$-$(2^n+1,13,\lambda)$ design as a linear combination of these seven $\lambda_B$ values with bounds given by the computed $m_{\lambda_B}$, providing a scalable, Burnside-based framework extendable to larger block sizes $k$. This approach significantly reduces computational complexity relative to direct enumeration and offers a clear path toward classifications for higher $k$.
Abstract
This paper investigates simple $3$-$(2^n+1,13,λ)$ designs admitting $\mathrm{PSL}$$(2,2^n)$ as an automorphism group. We determine all possible values of $λ$ by systematically analyzing the orbits of $13$-element subsets under the action of $\mathrm{PSL}$$(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.