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

Abstract Let k be a field. We are interested in the families of r -dimensional subspaces of k n with the following transversality property: any linear subspace of k n 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 .