Constructions are given of different kinds of flats in the projective space which are external to the Grassmannian of lines of PG(4,2). In particular it is shown that there exist precisely two GL(5,2)-orbits of external 4-flats, each with stabilizer group 31:5. (No 5-flat is external.) For each k=1,2,3, two distinct kinds of external k-flats are simply constructed out of certain partial spreads in PG(4,2) of size k+2. A third kind of external plane, with stabilizer 23:(7:3), is also shown to exist. With the aid of a certain key counting lemma, it is proved that the foregoing amounts to a complete classification of external flats.
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Shaw, R., Maks, JG., & Gordon, NA. (2005). The classification of flats in PG(9,2) which are external to the Grassmannian G1,4,2. Designs, Codes and Cryptography, 34(2-3), 203-227. https://doi.org/DOI: 10.1007/s10623-004-4855-6