Partial spreads in PG(4,2) and flats in PG(9,2) external to the Grassmannian G1,4,2

We consider the following 'even hyperplane construction' of flats in the projective space PG(9,2)=P(∧2V(5,2)) which are external to the Grassmannian G1,4,2 of lines of PG(4,2). Let the Grassmann image in G1,4,2 of a partial spread Sr={μ1,…,μr} in PG(4,2) be Cr={m1,…,mr}. Then Cr is an r-cap on G1,4,2. Using the recent classification [N.A. Gordon, R. Shaw, L.H. Soicher, Classification of Partial Spreads in PG(4,2), pp. 63, available from: http://www.hull.ac.uk/maths/people/rs/staffdetails.html] of partial spreads in PG(4,2), we determine those partial spreads Sr such that the projective space E(Cr)=〈Cr〉even generated by the r2 points mij≔mi+mj is an external flat. We show that, in this simple manner, we may construct seven out of the ten GL(5,2)-orbits of external flats.