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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/2512.24597 |
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| _version_ | 1866915701071020032 |
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| author | Kajiura, Hiroki Matsumoto, Makoto |
| author_facet | Kajiura, Hiroki Matsumoto, Makoto |
| contents | A difference set with parameters $(v, k, λ)$ is a subset $D$ of cardinality $k$ in a finite group $G$ of order $v$, such that the number $λ$ of occurrences of $g \in G$ as the ratio $d^{-1}d'$ in distinct pairs $(d, d')\in D\times D$ is independent of $g$. We prove the nonexistence of $(120, 35, 10)$-difference sets, which has been an open problem for 70 years since Bruck introduced the notion of nonabelian difference sets.
Our main tools are 1. a generalization of the category of finite groups to that of association schemes (actually, to that of relation partitions), 2. a generalization of difference sets to equi-distributed functions and its preservation by pushouts along quotients, 3. reduction to a linear programming in the nonnegative integer lattice with quadratic constraints. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A categorical proof of the nonexistence of (120, 35, 10)-difference sets Kajiura, Hiroki Matsumoto, Makoto Combinatorics 05B10 A difference set with parameters $(v, k, λ)$ is a subset $D$ of cardinality $k$ in a finite group $G$ of order $v$, such that the number $λ$ of occurrences of $g \in G$ as the ratio $d^{-1}d'$ in distinct pairs $(d, d')\in D\times D$ is independent of $g$. We prove the nonexistence of $(120, 35, 10)$-difference sets, which has been an open problem for 70 years since Bruck introduced the notion of nonabelian difference sets. Our main tools are 1. a generalization of the category of finite groups to that of association schemes (actually, to that of relation partitions), 2. a generalization of difference sets to equi-distributed functions and its preservation by pushouts along quotients, 3. reduction to a linear programming in the nonnegative integer lattice with quadratic constraints. |
| title | A categorical proof of the nonexistence of (120, 35, 10)-difference sets |
| topic | Combinatorics 05B10 |
| url | https://arxiv.org/abs/2512.24597 |