# The classification of flats in PG (9, 2) which are external to the grassmannian script G sign1,4,2

## Authors

Ron Shaw

Johannes G. Maks

##### 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 ofPG(4,2). In particular it is shown that there exist precisely twoGL(5,2)-orbits of external 4-flats, each with stabilizer group &cong;31:5. (No 5-flat is external.) For eachk=1,2,3, two distinct kinds of externalk-flats are simply constructed out of certain partial spreads inPG(4,2) of sizek+2. A third kind of external plane, with stabilizer &cong;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.

 Journal Article Type Article Publication Date 2005-02 Journal Designs, Codes, and Cryptography Print ISSN 0925-1022 Electronic ISSN 1573-7586 Publisher Springer Verlag Peer Reviewed Peer Reviewed Volume 34 Issue 2-3 Pages 203-227 Institution Citation Shaw, R., Maks, J. G., & Gordon, N. A. (2005). The classification of flats in PG (9, 2) which are external to the grassmannian script G sign1,4,2. Designs, codes, and cryptography, 34(2-3), 203-227. https://doi.org/10.1007/s10623-004-4855-6 DOI https://doi.org/10.1007/s10623-004-4855-6 Keywords Applied Mathematics; Computer Science Applications Publisher URL https://link.springer.com/article/10.1007%2Fs10623-004-4855-6