  An information-theoretic lower bound is developed for the caching system
studied by Maddah-Ali and Niesen. By comparing the proposed lower bound with
the decentralized coded caching scheme of Maddah-Ali and Niesen, the optimal
memory--rate tradeoff is characterized to within a multiplicative gap of $4.7$
for the worst case, improving the previous analytical gap of $12$. Furthermore,
for the case when users' requests follow the uniform distribution, the
multiplicative gap is tightened to $4.7$, improving the previous analytical gap
of $72$. As an independent result of interest, for the single-user average case
in which the user requests multiple files, it is proved that caching the most
requested files is optimal.
