Towards the Exact Memory Rate Tradeoff for the (4,5) Cache Network

The notion of coded caching was introduced by Maddah-Ali and Niesen when they demonstrated the utility of coding in caching systems. Since their seminal work, several schemes have been proposed to characterize optimal memory rate tradeoff to the caching problems. In this paper, we consider the (4, 5) cache network where the server has four files, each of size F bits, and five users are connected to the server through a common shared link. We consider the demands where each file in the server is requested by at least one user. For this cache network, we derive an improved lower bound for the small cache region, where the cache size in the range of $\displaystyle \frac{1}{5}F$ bits to $\displaystyle \frac{4}{5}F$ bits. We also introduce a new caching scheme to achieve the memory rate pair $\left(\displaystyle \frac{61}{20},\frac{1}{4}\right)$. We then derive a new lower bound for the cache region where cache size in the range of $\displaystyle \frac{61}{20}F$ bits to $\displaystyle \frac{16}{5}F$ bits to prove the optimality of the proposed scheme.