Symmetric $(15,8,4)$-designs in terms of the geometry of binary simplex codes of dimension $4$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913407175753728 |
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| author | Pankov, Mark Petelczyc, Krzysztof Żynel, Mariusz |
| author_facet | Pankov, Mark Petelczyc, Krzysztof Żynel, Mariusz |
| contents | Let $n=2^k-1$ and $m=2^{k-2}$ for a certain $k\ge 3$. Consider the point-line geometry of $2m$-element subsets of an $n$-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension $k$. For $k\ge 4$ the associated collinearity graph contains maximal cliques different from maximal singular subspaces. We investigate maximal cliques corresponding to symmetric $(n,2m,m)$-designs. The main results concern the case $k=4$ and give a geometric interpretation of the five well-known symmetric $(15,8,4)$-designs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19710 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetric $(15,8,4)$-designs in terms of the geometry of binary simplex codes of dimension $4$ Pankov, Mark Petelczyc, Krzysztof Żynel, Mariusz Combinatorics Let $n=2^k-1$ and $m=2^{k-2}$ for a certain $k\ge 3$. Consider the point-line geometry of $2m$-element subsets of an $n$-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension $k$. For $k\ge 4$ the associated collinearity graph contains maximal cliques different from maximal singular subspaces. We investigate maximal cliques corresponding to symmetric $(n,2m,m)$-designs. The main results concern the case $k=4$ and give a geometric interpretation of the five well-known symmetric $(15,8,4)$-designs. |
| title | Symmetric $(15,8,4)$-designs in terms of the geometry of binary simplex codes of dimension $4$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.19710 |