In this paper, in the first part of Section 7, D.J. Bernstein mentions that the original AKS paper (Lemma 4.4 and its proof) uses an extremely crude lower bound for the number of elements in $G$ (as it is defined in the original AKS paper). Bernstein gives the bound ${|T| + \deg h - 1 \choose |T|}$, which (barring the `$+1$' for the zero polynomial) is the same as the original AKS paper (which states ${l+d-1 \choose l}$, where $l$ equals $|T|$ and $d$ equals $\deg h$).
So the original AKS paper gives the same exact count for the number of elements in $G$. Why does Bernstein refer to the bound in the original AKS paper being crude? Or does he refer to the 'crude bound' being $(\frac{d}{l})^l$ as the bound for the binomial coefficient? If so, I don't exactly see how Bernsteins discussion in this part determines a better/exact lower bound on $G$.
Thanks in advance for any clarification.