A gap result for Cameron–Liebler $k$-classes

Abstract

The notion of Cameron–Liebler line classes was generalized in Rodgers et al. (0000) to Cameron–Liebler $k$-classes, where $k=1$ corresponds to the line classes. Such a set consists of $x{\left[\genfrac{}{}{0ex}{}{2k+1}{k}\right]}_q$ subspaces of dimension $k$ in $\mathrm{PG}(2k+1,q)$ where $k\ge 1$ and $x\ge 0$ are integers such that every regular $k$-spread of $\mathrm{PG}(2k+1,q)$ contains exactly $x$ subspaces from the set. Examples are known for $x\le 2$. The authors of Rodgers et al. (0000) show that there are no Cameron–Liebler $k$-classes when $k=2$ and $3\le x\le \sqrt{q}$, or when $3\le k\le qlogq-q$ and $3\le x\le \sqrt[3]{q∕2}$. We improve these results by weakening the condition on the upper bound for $x$ to a bound that is linear in $q$. For this, we use a technique that was originally used to extend nets to affine planes.