Diameter Bound for Finite Simple Groups of Large Rank

Given a non-abelian finite simple group $G$ of Lie type, and an arbitrary generating set $S$, it is conjectured by Laszlo Babai that its Cayley graph $\mathrm{\Gamma }(G,S)$ will have a diameter of $(\log \mathrm{\mid }G\mathrm{\mid }{)}^{O(1)}$. However, little progress has been made when the rank of $G$ is large. In this article, we shall show that if $G$ has rank $n$, and its base field has order $q$, then the diameter of $\mathrm{\Gamma }(G,S)$ would be $q^{O(n(\log n+\log q{)}^3)}$.