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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.20614 |
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| _version_ | 1866913451808391168 |
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| author | Shan, Yunjing Zhou, Junling |
| author_facet | Shan, Yunjing Zhou, Junling |
| contents | There is a large literature on cover-free families of finite sets, because of their many applications in combinatorial group testing, cryptographic and communications. This work studies the generalization of cover-free families from sets to finite vector spaces. Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$ and let $\left[V\atop k\right]_q$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq \left[V\atop k\right]_q$ is called cover-free if there are no three distinct subspaces $F_{0}, F_{1}, F_{2}\in \mathcal{F}$ such that $F_{0}\leq (F_{0}\cap F_{1})+(F_{0}\cap F_{2})$. A family $\mathcal{H}\subseteq \left[V\atop k\right]_q$ is called a $q$-Steiner system $S_{q}(t, k, n)$ if for every $T\in \left[V\atop t\right]_q$, there is exactly one $H\in \mathcal{H}$ such that $T\leq H$. In this paper we investigate cover-free families in the vector space $V$. Firstly, we determine the maximum size of a cover-free family in $\left[V\atop k\right]_q$. Secondly, we characterize the structures of all maximum cover-free families which are closely related to $q$-Steiner systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20614 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On cover-free families of finite vector spaces Shan, Yunjing Zhou, Junling Combinatorics There is a large literature on cover-free families of finite sets, because of their many applications in combinatorial group testing, cryptographic and communications. This work studies the generalization of cover-free families from sets to finite vector spaces. Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$ and let $\left[V\atop k\right]_q$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq \left[V\atop k\right]_q$ is called cover-free if there are no three distinct subspaces $F_{0}, F_{1}, F_{2}\in \mathcal{F}$ such that $F_{0}\leq (F_{0}\cap F_{1})+(F_{0}\cap F_{2})$. A family $\mathcal{H}\subseteq \left[V\atop k\right]_q$ is called a $q$-Steiner system $S_{q}(t, k, n)$ if for every $T\in \left[V\atop t\right]_q$, there is exactly one $H\in \mathcal{H}$ such that $T\leq H$. In this paper we investigate cover-free families in the vector space $V$. Firstly, we determine the maximum size of a cover-free family in $\left[V\atop k\right]_q$. Secondly, we characterize the structures of all maximum cover-free families which are closely related to $q$-Steiner systems. |
| title | On cover-free families of finite vector spaces |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.20614 |