Higgledy-piggledy sets in projective spaces of small dimension

TL;DR

This paper studies special sets of subspaces in small-dimensional projective spaces that are well-distributed, introduces construction methods, and explores their properties and applications in coding and graph theory.

Contribution

It presents new constructions and existence results for higgledy-piggledy sets of subspaces in projective spaces of small dimension, including explicit examples and their applications.

Findings

Existence of six lines in higgledy-piggledy arrangement in PG(4,q)

Existence of six planes with specific intersections in PG(4,q)

Existence of two sets of planes and solids in PG(5,q) in higgledy-piggledy arrangement

Abstract

This work focuses on higgledy-piggledy sets of $k$-subspaces in $\text{PG}(N,q)$, i.e. sets of projective subspaces that are 'well-spread-out'. More precisely, the set of intersection points of these $k$-subspaces with any $(N-k)$-subspace $\kappa$ of $\text{PG}(N,q)$ spans $\kappa$ itself. We highlight three methods to construct small higgledy-piggledy sets of $k$-subspaces and discuss, for $k\in\{1,N-2\}$, 'optimal' sets that cover the smallest possible number of points. Furthermore, we investigate small non-trivial higgledy-piggledy sets in $\text{PG}(N,q)$, $N\leqslant5$. Our main result is the existence of six lines of $\text{PG}(4,q)$ in higgledy-piggledy 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 higgledy-piggledy arrangement, two of which maximally intersect, as well as the existence of two higgledy-piggledy sets in $\text{PG}(5,q)$ consisting of eight planes and seven solids, respectively. Finally, we translate these geometrical results to a coding- and graph-theoretical context.