Bounding the parameter $\beta$ of a distance-regular graph with classical parameters

Let $\mathrm{\Gamma }$ be a distance-regular graph with classical parameters $(D,b,\alpha ,\beta )$ satisfying $b\ge 2$ and $D\ge 3$. Let $r=1+b+b^2+\cdots +b^{D-1}$. In 1999, K. Metsch showed that there exists a positive constant $C(\alpha ,b)$ only depending on $\alpha$ and $b$, such that if $\beta \ge C(\alpha ,b)r^2$, then either $\mathrm{\Gamma }$ is a Grassmann graph or a bilinear forms graph. In this work, we show that for $b\ge 2$ and $D\ge 3$, then there exists a constant $C_1(\alpha ,b)$ only depending on $\alpha$ and $b$, such that if $\beta \ge C_1(\alpha ,b)r$, then either $\mathrm{\Gamma }$ is a Grassmann graph, or a bilinear forms graph.