The Exact Rate Memory Tradeoff for Small Caches with Coded Placement

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 ≥ 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 ≤ 1 K . In this paper we focus on small caches where M ∈ [ 0, N K ] and derive new lower bounds. For the case when ⌈ K+1 2 ⌉ ≤ N ≤ K and M ∈ [ 1 K , N K (N−1) ] , our lower bounds demonstrate that the caching scheme introduced by Gómez-Vilardebó is optimal and thus extend the characterization of the exact rate memory tradeoff. For the case 1 ≤ N ≤ ⌈ K+1 2 ⌉ , we show that the new lower bounds improve upon the previously known lower bounds. Index Terms Coded caching, coded placement, exact rate memory tradeoff, lower bounds.