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

TL;DR

This paper determines the dimension of the space generated by characteristic vectors of $q$-Steiner systems, generalizing previous results on ordinary Steiner systems to the $q$-analog setting.

Contribution

It proves that the dimension equals ${nrack k}_{q}-{nrack t}_{q}+1$ when such $q$-Steiner systems exist, extending Ghodrati's 2019 work.

Findings

The dimension of the space is explicitly computed for $q$-Steiner systems.

The result generalizes known results from ordinary Steiner systems to the $q$-analog case.

The formula applies when at least one $q$-Steiner system exists.

Abstract

Fix a prime power $q$ and parameters $1\leq t\leq k\leq n$, 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 ${n\brack k}_{q}-{n\brack t}_{q}+1$. This generalizes the 2019 work of Ghodrati \cite{ghodrati2019dimension} on ordinary Steiner systems.