Almost Steiner systems in finite classical polar spaces

Abstract

Let $Q$ be a finite classical polar space with rank n and $P$ be a collection of k-dimensional subspaces in $Q$ called blocks. Let Λ be a set of nonnegative integers. A pair $(Q,P)$ is called a t-${(n,k,\Lambda )}_Q$ design if every t-dimensional subspace in $Q$ is contained in exactly $\lambda \in \Lambda$ blocks of $P$. A t-${(n,k,1)}_Q$ design is called a Q-Steiner system, which has ${\left[\begin{array}{c}n\\ t\end{array}\right]}_Q/{\left[\begin{array}{c}k\\ t\end{array}\right]}_q$ blocks. Despite knowing very little about the existence of Q-Steiner systems, we demonstrate that given any positive integers k and t satisfying $k>t$, for any finite polar space $Q$ with a sufficiently large rank n, there exists a t-${(n,k,\{1,2\left\}\right)}_Q$ design with $(1+o(1\left)\right){\left[\begin{array}{c}n\\ t\end{array}\right]}_Q/{\left[\begin{array}{c}k\\ t\end{array}\right]}_q$ blocks.