Dual incidences and t-designs in vector spaces

Abstract

Let $V$ be an $n$-dimensional vector space over $F_q$ and $H$ is any set of $k$-dimensional subspaces of $V$. We construct two incidence structures $D_{max}\left(H\right)$ and $D_{min}\left(H\right)$ using subspaces from $H$. The points are subspaces from $H$. The blocks of $D_{max}\left(H\right)$ are indexed by all hyperplanes of $V$, while the blocks of $D_{min}\left(H\right)$ are indexed by all subspaces of dimension 1. We show that $D_{max}\left(H\right)$ and $D_{min}\left(H\right)$ are dual in the sense that their incidence matrices are dependent, one can be calculated from the other. Additionally, if $H$ is a $t-{\left(n,k,\lambda \right)}_q$-design we prove new matrix equations for incidence matrices of $D_{max}\left(H\right)$ and $D_{min}\left(H\right)$.