Higgledypiggledy sets in projective spaces of small dimension
Abstract
This work focuses on higgledypiggledy sets of $k$subspaces in $\text{PG}(N,q)$, i.e. sets of projective subspaces that are 'wellspreadout'. More precisely, the set of intersection points of these $k$subspaces with any $(Nk)$subspace $\kappa$ of $\text{PG}(N,q)$ spans $\kappa$ itself. We highlight three methods to construct small higgledypiggledy sets of $k$subspaces and discuss, for $k\in\{1,N2\}$, 'optimal' sets that cover the smallest possible number of points. Furthermore, we investigate small nontrivial higgledypiggledy sets in $\text{PG}(N,q)$, $N\leqslant5$. Our main result is the existence of six lines of $\text{PG}(4,q)$ in higgledypiggledy arrangement, two of which intersect. Exploiting the construction methods mentioned above, we also show the existence of six planes of $\text{PG}(4,q)$ in higgledypiggledy arrangement, two of which maximally intersect, as well as the existence of two higgledypiggledy sets in $\text{PG}(5,q)$ consisting of eight planes and seven solids, respectively. Finally, we translate these geometrical results to a coding and graphtheoretical context.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.08572
 Bibcode:
 2021arXiv210908572D
 Keywords:

 Mathematics  Combinatorics;
 05B25;
 94B05;
 51E20;
 51E21
 EPrint:
 21 pages, 1 figure