Projective de Bruijn Sequences

Let F, denote the q-element field. A q-ary de Bruijn sequence of degree m is a cyclic sequence of elements of F-q, such that every element in F-q(m) appears exactly once as a consecutive m-tuple in the cyclic sequence. We consider its projective analogue; namely, a cyclic sequence such that every point in the projective space (F-q(m+1) - {0})/(F-q(x)) appears exactly once as a consecutive (m + 1)-tuple. We have an explicit formula (q!) q(m) - 1/q-1 q(-m) for the number of distinct such sequences.