A Note on Fano Configurations in the Projective Space PG(5,2)

Abstract

Let $n\geq $ $2$ and let $U_{j}\mid j\in J$, with $|J|=n^{2}+n+1$, be a set of disjoint subspaces (of the same dimension) of some finite projective space $PG(N,q)$ with the property that the number of such subspaces in the span of any two such subspaces is always $n+1$ and the intersection of any two different such spans is always a subspace $U_{j}$ (so we have a projective plane of order $n$ with point set $U_{j}\mid j\in J.$) In this work we search for Fano configurations in PG(5,2) whose lines are 3-spaces and points are lines.