Almost affinely disjoint subspaces

In this work, we introduce a natural notion concerning finite vector spaces. A family of k-dimensional subspaces of F_qⁿ, which forms a partial spread, 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.