Almost affinely disjoint subspaces

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