| Abstrakt: |
Let V $V$ be an n $n$‐dimensional vector space over Fq ${{\mathbb{F}}}_{q}$ and H ${\rm{ {\mathcal H} }}$ is any set of k $k$‐dimensional subspaces of V $V$. We construct two incidence structures Dmax(H) ${{\mathscr{D}}}_{max}({\rm{ {\mathcal H} }})$ and Dmin(H) ${{\mathscr{D}}}_{min}({\rm{ {\mathcal H} }})$ using subspaces from H ${\rm{ {\mathcal H} }}$. The points are subspaces from H ${\rm{ {\mathcal H} }}$. The blocks of Dmax(H) ${{\mathscr{D}}}_{max}({\rm{ {\mathcal H} }})$ are indexed by all hyperplanes of V $V$, while the blocks of Dmin(H) ${{\mathscr{D}}}_{min}({\rm{ {\mathcal H} }})$ are indexed by all subspaces of dimension 1. We show that Dmax(H) ${{\mathscr{D}}}_{max}({\rm{ {\mathcal H} }})$ and Dmin(H) ${{\mathscr{D}}}_{min}({\rm{ {\mathcal H} }})$ are dual in the sense that their incidence matrices are dependent, one can be calculated from the other. Additionally, if H ${\rm{ {\mathcal H} }}$ is a t−(n,k,λ)q $t-{(n,k,\lambda)}_{q}$‐design we prove new matrix equations for incidence matrices of Dmax(H) ${{\mathscr{D}}}_{max}({\rm{ {\mathcal H} }})$ and Dmin(H) ${{\mathscr{D}}}_{min}({\rm{ {\mathcal H} }})$. [ABSTRACT FROM AUTHOR] |
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