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For a finite group   G  , letdiam   (  G  )    denote the maximum diameter of a connected Cayley graph of   G  . A well‐known conjecture of Babai states thatdiam   (  G  )    is bounded by(   log  2   |  G  |  )    O  (  1  )in case   G   is a non‐abelian finite simple group. Let   G   be a finite simple group of Lie type of Lie rank   n   over the field    F  q   . Babai's conjecture has been verified in case   n   is bounded, but it is wide open in case   n   is unbounded. Recently, Biswas and Yang proved thatdiam   (  G  )    is bounded by    q   O  (  n    (   log  2   n  +   log  2   q  )   3   )    . We show that in factdiam    (  G  )   <   q   O  (  n    (   log  2   n  )   2   )holds. Note that our bound is significantly smaller than the order of   G   for   n   large, even if   q   is large. As an application, we show that more generallydiam    (  H  )   <   q   O  (  n    (   log  2   n  )   2   )holds for any subgroup   H   ofGL   (  V  )   , where   V   is a vector space of dimension   n   defined over the field    F  q   .

An improved diameter bound for finite simple groups of Lie type