Almost Affinely Disjoint Subspaces

Abstract

In this work, we introduce a natural notion concerning vector finite spaces. A family of k-dimensional subspaces of F n q is called almost affinely disjoint if any (k + 1)-dimensional subspace containing a subspace from the family non-trivially intersects with only a few subspaces from the family. The central question discussed in the paper is the polynomial growth (in q) of the maximal cardinality of these families given the parameters k and n. For the cases k = 1 and k = 2, optimal families are constructed. For other settings, we find lower and upper bounds on the polynomial growth. Additionally, some connections with problems in coding theory are shown.