  The idea of coded caching for content distribution networks was introduced by
Maddah-Ali and Niesen, who considered the canonical $(N, K)$ cache network in
which a server with $N$ files satisfy the demands of $K$ users (equipped with
independent caches of size $M$ each). Among other results, their work provided
a characterization of the exact rate memory tradeoff for the problem when
$M\geq\frac{N}{K}(K-1)$. In this paper, we improve this result for large caches
with $M\geq \frac{N}{K}(K-2)$. For the case
$\big\lceil\frac{K+1}{2}\big\rceil\leq N \leq K$, we propose a new coded
caching scheme, and derive a matching lower bound to show that the proposed
scheme is optimal. This extends the characterization of the exact rate memory
tradeoff to the case $M\geq \frac{N}{K}\Big(K-2+\frac{(K-2+1/N)}{(K-1)}\Big)$.
For the case $1\leq N\leq \big\lceil\frac{K+1}{2}\big\rceil$, we derive a new
lower bound, which demonstrates that the scheme proposed by Yu et al. is
optimal and thus extend the characterization of the exact rate memory tradeoff
to the case $M\geq \frac{N}{K}(K-2)$.
