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arXiv · 2301.13001

On the minimum size of linear sets

Sam Adriaensen·Paolo Santonastaso
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Abstract

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 $\pi$ 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 $\pi$ is a hyperplane, and in the case that $\pi$ is a lower dimensional subspace.

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Sam Adriaensen, Paolo Santonastaso. 2023-01-30. On the minimum size of linear sets. https://doi.org/10.37236/12424

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