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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2510.25360 |
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| _version_ | 1866915585382678528 |
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| author | Galici, Mario Montinaro, Alessandro |
| author_facet | Galici, Mario Montinaro, Alessandro |
| contents | A complete classification of the flag-transitive point-imprimitive symmetric $2$-$(v,k,λ)$ designs with $v<100$ is provided. Apart from the known examples with $λ\leq 10$, the complementary design of $PG_{5}(2)$, and the $2$-design $\mathcal{S}^{-}(3)$ constructed by Kantor in \cite{Ka75}, we found two non isomorphic $2$-$(64,28,12)$ designs. They were constructed via computer as developments of $(64,28,12)$-difference sets by AbuGhneim in \cite{OAG}. In the present paper, independently from \cite{OAG}, we construct the aforementioned two $2$-designs and we prove that their full automorhpism group is flag-transitive and point-imprimitive. The construction is theoretical and relies on the the absolutely irreducible $8$-dimensional $\mathbb{F}_{2}$-representation of $PSL_{2}(7)$. Our result, together with that about the flag-transitive point-primitive symmetric $2$-designs with $v<2500$ by Braić-Golemac-Mandić-Vučičić \cite{BGMV}, provides a complete classification of the flag-transitive $2$-designs with $v<100$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25360 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Flag-Transitive and Point-Imprimitive Symmetric $(v,k,λ)$ Designs with $v<100$ Galici, Mario Montinaro, Alessandro Group Theory A complete classification of the flag-transitive point-imprimitive symmetric $2$-$(v,k,λ)$ designs with $v<100$ is provided. Apart from the known examples with $λ\leq 10$, the complementary design of $PG_{5}(2)$, and the $2$-design $\mathcal{S}^{-}(3)$ constructed by Kantor in \cite{Ka75}, we found two non isomorphic $2$-$(64,28,12)$ designs. They were constructed via computer as developments of $(64,28,12)$-difference sets by AbuGhneim in \cite{OAG}. In the present paper, independently from \cite{OAG}, we construct the aforementioned two $2$-designs and we prove that their full automorhpism group is flag-transitive and point-imprimitive. The construction is theoretical and relies on the the absolutely irreducible $8$-dimensional $\mathbb{F}_{2}$-representation of $PSL_{2}(7)$. Our result, together with that about the flag-transitive point-primitive symmetric $2$-designs with $v<2500$ by Braić-Golemac-Mandić-Vučičić \cite{BGMV}, provides a complete classification of the flag-transitive $2$-designs with $v<100$. |
| title | The Flag-Transitive and Point-Imprimitive Symmetric $(v,k,λ)$ Designs with $v<100$ |
| topic | Group Theory |
| url | https://arxiv.org/abs/2510.25360 |