  The idea of coded caching was introduced by Maddah-Ali and Niesen who
demonstrated the advantages of coding in caching problems. To capture the
essence of the problem, they introduced the $(N, K)$ canonical cache network in
which $K$ users with independent caches of size $M$ request files from a server
that has $N$ files. Among other results, the caching scheme and lower bounds
proposed by them led to a characterization of the exact rate memory tradeoff
when $M\geq \frac{N}{K}(K-1)$. These lower bounds along with the caching scheme
proposed by Chen et al. led to a characterization of the exact rate memory
tradeoff when $M\leq \frac{1}{K}$. In this paper we focus on small caches where
$M\in \left[0,\frac{N}{K}\right]$ and derive new lower bounds. For the case
when $\big\lceil\frac{K+1}{2}\big\rceil\leq N \leq K$ and $M\in
\big[\frac{1}{K},\frac{N}{K(N-1)}\big]$, our lower bounds demonstrate that the
caching scheme introduced by G{\'o}mez-Vilardeb{\'o} is optimal and thus extend
the characterization of the exact rate memory tradeoff. For the case $1\leq
N\leq \big\lceil\frac{K+1}{2}\big\rceil$, we show that the new lower bounds
improve upon the previously known lower bounds.
