Babai, Kantor and Lubbotsky proved the following theorem. THEOREM 1.1. There is a constant $C$ such that every nonabelian finite simple group $G$ has a set $S$ of at most 7 generators for which the diameter of $Cay(G,S)$ is at most $C\log|G|$. Then they remark that "A crude estimate for $C$ is $10^{10}$, but we will not include the bookkeeping required to estimate $C$." This is my question. "Is there a finite simple group $G$ for which there exists a generating set $S$ which satisfies the conditions in the above theorem for some reasonably small $C$ (comparing to the order of $G$)?"