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The Classification of Flats in \({\boldsymbol {PG}}({\bf 9,2})\) which are External to the Grassmannian \({\cal G}_{\bf 1,4,2}\)

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Abstract

Constructions are given of different kinds of flats in the projective space \(PG(9,2)={\mathbb P}(\wedge^{2}V(5,2))\) which are external to the Grassmannian \({\cal G}_{\bf 1,4,2}\) 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, J.G. & Gordon, N.A. The Classification of Flats in \({\boldsymbol {PG}}({\bf 9,2})\) which are External to the Grassmannian \({\cal G}_{\bf 1,4,2}\). Des Codes Crypt 34, 203–227 (2005). https://doi.org/10.1007/s10623-004-4855-6

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