On the construction of a family of transversal subspaces over finite fields

Let k be a field. We are interested in the families of r-dimensional subspaces of kn with the following transversality property: any linear subspace of kn of dimension n-r is transversal to at least one element of the family. While it is known how to build such families in polynomial time over infinite fields k, no such technique is known for finite fields. However, transversal families in dimension n can be built when the field k is large enough with respect to n. We improve here on how large k needs to be with respect to the considered dimension n.