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

Autor: Neil Gordon, Ron Shaw, Johannes G. Maks
Rok vydání: 2005
Předmět:
Zdroj: Designs, Codes and Cryptography. 34:203-227
ISSN: 1573-7586
0925-1022
Popis: 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.
Databáze: OpenAIRE
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