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arXiv · 2503.16050

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

Takara Kondo·Yuto Nogata
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Abstract

This paper investigates simple $3$-$(2^n+1,13,\lambda)$ designs admitting $\mathrm{PSL}$$(2,2^n)$ as an automorphism group. We determine all possible values of $\lambda$ by systematically analyzing the orbits of $13$-element subsets under the action of $\mathrm{PSL}$$(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.

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Takara Kondo, Yuto Nogata. 2025-03-20. Simple $3$-designs of $\mathrm{PSL}(2,2^n)$ with block size $13$. https://arxiv.org/abs/2503.16050

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