On the minimum size of linear sets
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910001235230720 |
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| author | Adriaensen, Sam Santonastaso, Paolo |
| author_facet | Adriaensen, Sam Santonastaso, Paolo |
| contents | Recently, a lower bound was established on the size of linear sets in projective spaces, that intersect a hyperplane in a canonical subgeometry. There are several constructions showing that this bound is tight. In this paper, we generalize this bound to linear sets meeting some subspace $π$ in a canonical subgeometry. We obtain a tight lower bound on the size of any $\mathbb F_q$-linear set spanning $\text{PG}(d,q^n)$ in case that $n \leq q$ and $n$ is prime. We also give constructions of linear sets attaining equality in the former bound, both in the case that $π$ is a hyperplane, and in the case that $π$ is a lower dimensional subspace. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_13001 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the minimum size of linear sets Adriaensen, Sam Santonastaso, Paolo Combinatorics 51E20, 05B25 Recently, a lower bound was established on the size of linear sets in projective spaces, that intersect a hyperplane in a canonical subgeometry. There are several constructions showing that this bound is tight. In this paper, we generalize this bound to linear sets meeting some subspace $π$ in a canonical subgeometry. We obtain a tight lower bound on the size of any $\mathbb F_q$-linear set spanning $\text{PG}(d,q^n)$ in case that $n \leq q$ and $n$ is prime. We also give constructions of linear sets attaining equality in the former bound, both in the case that $π$ is a hyperplane, and in the case that $π$ is a lower dimensional subspace. |
| title | On the minimum size of linear sets |
| topic | Combinatorics 51E20, 05B25 |
| url | https://arxiv.org/abs/2301.13001 |