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.

翻译：Maddah- Ali 和 Niesen 介绍了编码缓存的概念。 Maddah- Ali 和 Niesen 展示了在缓存问题中编码的优点。 为了捕捉问题的实质, 他们引入了 $( N, K) 的 Canonic 缓存网络, 其中, K$ 用户拥有独立的大小的缓存 $M$ 请求文件 $M 。 除其他结果外, 由他们提议的缓存办法和下限导致对精确速内存交易的描述, 当 $M\ qqqqq {\ k} (K-1) 和 美元( K-1) (K-1) (K-1) 美元时, 这些较低的边框加上Chen 等人提出的缓存计划, 导致对 $( N, K) 和 N\\\\ k\ k\\ k\ k} 美元 美元 的准确的缓存交易的准确率的定性。 显示 G\\\\\\ k\ k\\\ k\ k\ k\\\\\\\\\\\ lifr) lix 新的 交易中的 lide srel lade lax lax 系统, lax lax 。 lax lax lax lax g\ g\ g\ g\\\\ k\\\\\\\\\\\ k\\\\ k\ k\\\\\\ k\ k\ k\\ k\\\\\\\\\\\\\\\ k\ k\ k\\ k\\\\\\ k\ k\ k\ k\ g\ k\ k\ k\\ k\ k\ k\ g\ g\ k\ k\ g\ g\ g\ k\ g\ g\ k\ k\ g\ k\ k\ g\ g\ g\ g\ k\ g\ crudecredec) lax lax lax lax lax lax lax lax lax lax lax lax lax lax lax lax lax lax lax lax lax la