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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.06369 |
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| _version_ | 1866910198619176960 |
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| author | Li, Qilong Weiß, Charlene Zhou, Yue |
| author_facet | Li, Qilong Weiß, Charlene Zhou, Yue |
| contents | Fix a prime power $q$ and parameters $1\leq t\leq k\leq n$, the corresponding Steiner system in the Grassmann scheme, or the $q$-Steiner system, is a collection $\mathfrak{B}$ of $k$-dimensional subspaces of $\mathbb{F}_{q}^n$ such that for each $t$-dimensional subspace $T$, there exists exactly one element of $\mathfrak{B}$ containing $T$. The dimension of Steiner systems in the Grassmann scheme is defined to be the dimension of the $\mathbb{Q}$-vector space spanned by the characteristic vectors of all these $q$-Steiner systems. In this paper, we prove that when a quadruple $(t,k,n,q)$ admits at least one $q$-Steiner system, the corresponding dimension is equal to ${n\brack k}_{q}-{n\brack t}_{q}+1$. This generalizes the 2019 work of Ghodrati \cite{ghodrati2019dimension} on ordinary Steiner systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06369 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the dimension of the space generated by characteristic vectors of $q$-Steiner systems Li, Qilong Weiß, Charlene Zhou, Yue Combinatorics 05B05, 51E10 Fix a prime power $q$ and parameters $1\leq t\leq k\leq n$, the corresponding Steiner system in the Grassmann scheme, or the $q$-Steiner system, is a collection $\mathfrak{B}$ of $k$-dimensional subspaces of $\mathbb{F}_{q}^n$ such that for each $t$-dimensional subspace $T$, there exists exactly one element of $\mathfrak{B}$ containing $T$. The dimension of Steiner systems in the Grassmann scheme is defined to be the dimension of the $\mathbb{Q}$-vector space spanned by the characteristic vectors of all these $q$-Steiner systems. In this paper, we prove that when a quadruple $(t,k,n,q)$ admits at least one $q$-Steiner system, the corresponding dimension is equal to ${n\brack k}_{q}-{n\brack t}_{q}+1$. This generalizes the 2019 work of Ghodrati \cite{ghodrati2019dimension} on ordinary Steiner systems. |
| title | On the dimension of the space generated by characteristic vectors of $q$-Steiner systems |
| topic | Combinatorics 05B05, 51E10 |
| url | https://arxiv.org/abs/2605.06369 |