Points and Lines

Split buildings of type $$\mathsf {F_4}$$ F 4 in buildings of type $$\mathsf {E_6}$$ E 6

“…For this reason, we extend the equator geometry to the "extended equator geometry", which is indeed a polar space of type B 4 , as we will show in Proposition 5. 18. We further show that projective subspaces of Γ all of whose lines are hyperbolic lines and which are at most 3-dimensional, are contained in an extended equator geometry (Lemma 5.21).…”

“…-Parabolic quadrics or polar spaces of type B n to mean that the given point-line geometry conforms to a building of type B n where points and lines are the elements of type 1 and 2, respectively; -Symplectic polar spaces, or polar spaces of type C n , which conform to buildings of type C n ; -Quadrics of type D n are hyperbolic quadrics whose oriflamme complexes are buildings of type D n ; -Finally, a metasymplectic space is a point-line geometry (a so-called parapolar space, see [18]) associated with a (thick) building of type F 4 .…”

On the equivalence of certain quasi-Hermitian varieties

“…Let k be the unique solution of (11). Since k = l + νm, we find q 2 pairs (l, m) satisfying (10) and q possible values of n for each of these pairs (l, m) because of ( 9). Thus we obtain that the number of affine lines through the point P and contained in B α,β is q 2 q/q 2 = q.…”

On the Grassmann graph of linear codes

“…In general, we say that a subgraph is isometrically embedded in a larger graph if there exists a distance-preserving map among them (see also Definition 2.3); see also [5].…”