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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.16050 |
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| _version_ | 1866915206513295360 |
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| author | Kondo, Takara Nogata, Yuto |
| author_facet | Kondo, Takara Nogata, Yuto |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16050 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simple $3$-designs of $\mathrm{PSL}(2,2^n)$ with block size $13$ Kondo, Takara Nogata, Yuto Combinatorics 05B05(Primary), 20G40(Secondary) 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. |
| title | Simple $3$-designs of $\mathrm{PSL}(2,2^n)$ with block size $13$ |
| topic | Combinatorics 05B05(Primary), 20G40(Secondary) |
| url | https://arxiv.org/abs/2503.16050 |