On the dimension of the space generated by characteristic vectors of $q$-Steiner systems

Abstract

Fix a prime power $q$ and parameters $1\le t\le k\le n$, the corresponding Steiner system in the Grassmann scheme, or the $q$-Steiner system, is a collection $\mathfrak{B}$ of $k$-dimensional subspaces of ${\mathbb{F}}_q^n$ such that for each $t$-dimensional subspace $T$, there exists exactly one element of $\mathfrak{B}$ containing $T$. The dimension of Steiner systems in the Grassmann scheme is defined to be the dimension of the $\mathbb{Q}$-vector space spanned by the characteristic vectors of all these $q$-Steiner systems. In this paper, we prove that when a quadruple $(t,k,n,q)$ admits at least one $q$-Steiner system, the corresponding dimension is equal to ${\left[\genfrac{}{}{0ex}{}{n}{k}\right]}_q-{\left[\genfrac{}{}{0ex}{}{n}{t}\right]}_q+1$. This generalizes the 2019 work of Ghodrati \cite{ghodrati2019dimension} on ordinary Steiner systems.