The q-analog of the middle levels problem

The well-known middle levels problem is to find a Hamiltonian cycle in the graph induced from the binary Hamming graph H2(2k+1) by the words of weight k or k+1. In this paper we define the q-analog of the middle levels problem. Let n=2k+1 and let  q  be a power of a prime number. Consider the set of (k+1)-dimensional subspaces and the set of k-dimensional subspaces of Fqn. Can these subspaces be ordered in a way that for any two adjacent subspaces X and Y, either X⊂Y or Y⊂X? A construction method which yields many Hamiltonian cycles for any given q and k=2 is presented.