Recently, a lower bound on the size of linear sets in projective spaces intersecting a hyperplane in a canonical subgeometry was established. 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 $\mathrm{PG}(d,q^n)$ 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.
@article{10_37236_12424,
author = {Sam Adriaensen and Paolo Santonastaso},
title = {On the minimum size of linear sets},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/12424},
zbl = {7975069},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12424/}
}
TY - JOUR
AU - Sam Adriaensen
AU - Paolo Santonastaso
TI - On the minimum size of linear sets
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/12424/
DO - 10.37236/12424
ID - 10_37236_12424
ER -
%0 Journal Article
%A Sam Adriaensen
%A Paolo Santonastaso
%T On the minimum size of linear sets
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/12424/
%R 10.37236/12424
%F 10_37236_12424
Sam Adriaensen; Paolo Santonastaso. On the minimum size of linear sets. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12424
