The polynomial degree of the Grassmannian G1,n,2

For a subset psi of PG(N, 2) a known result states that psi has polynomial degree <= r, r <= N, if and only if psi intersects every r-flat of PG(N, 2) in an odd number of points. Certain refinements of this result are considered, and are then applied in the case when psi is the Grassmannian G (l, n, 2) subset of PG (N, 2), N = ((n+ 1)(2)) -1, to show that for n < 8 the polynomial degree of G (l, n, 2) is ((n)(2))- 1.