Existence of $S(2,9,369)$, new unitals of order $6$ and other Steiner systems with block length $\ge 7$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917301504180224 |
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| author | Hetman, Ivan |
| author_facet | Hetman, Ivan |
| contents | Whereas Steiner systems $S(2,k,v)$ with block length $k \le 5$ have large amount of examples and the existence is established for all admissible $v$, for $k\ge 6$ only few examples are known even for decided cases. In this paper the existence of $S(2,9,369)$ is established and some new examples for other admissible pairs $(k,v)$ are given. In particular, lots of new unitals of order $6$ (or $S(2,7,217)$) together with $S(2,7,175)$, $S(2,7,259)$, $S(2,8,120)$, $S(2,8,504)$, $S(2,9,513)$ are presented. Found examples suggest two conjectures on infinite series of designs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of $S(2,9,369)$, new unitals of order $6$ and other Steiner systems with block length $\ge 7$ Hetman, Ivan Combinatorics 51E10 (Primary), 51E05 (Secondary) Whereas Steiner systems $S(2,k,v)$ with block length $k \le 5$ have large amount of examples and the existence is established for all admissible $v$, for $k\ge 6$ only few examples are known even for decided cases. In this paper the existence of $S(2,9,369)$ is established and some new examples for other admissible pairs $(k,v)$ are given. In particular, lots of new unitals of order $6$ (or $S(2,7,217)$) together with $S(2,7,175)$, $S(2,7,259)$, $S(2,8,120)$, $S(2,8,504)$, $S(2,9,513)$ are presented. Found examples suggest two conjectures on infinite series of designs. |
| title | Existence of $S(2,9,369)$, new unitals of order $6$ and other Steiner systems with block length $\ge 7$ |
| topic | Combinatorics 51E10 (Primary), 51E05 (Secondary) |
| url | https://arxiv.org/abs/2511.20708 |