Higgledy-Piggledy Sets in Projective Spaces of Small Dimension

This work focuses on higgledy-piggledy sets of k-subspaces in 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 κ of PG(N,q) spans κ itself. We highlight three methods to construct small higgledy-piggledy sets of k-subspaces and discuss, for k∈\1,N-2\, 'optimal' sets that cover the smallest possible number of points. Furthermore, we investigate small non-trivial higgledy-piggledy sets in PG(N,q), N5. Our main result is the existence of six lines of 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 PG(4,q) in higgledy-piggledy arrangement, two of which maximally intersect, as well as the existence of two higgledy-piggledy sets in PG(5,q) consisting of eight planes and seven solids, respectively. Finally, we translate these geometrical results to a coding- and graph-theoretical context.