Higgledy-piggledy sets in projective spaces of small dimension

Abstract

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), N 5. 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.