We consider a coded caching network consisting of a server with a library of $N$ files connected to $K$ users, where each user is equipped with a dedicated cache of size $M_p$ units. In addition to that, the network consists of $\Lambda\leq K$ helper caches, each with a size $M_h$ units. Each helper cache can serve an arbitrary number of users; however, each user can access only a single helper cache. Also, we assume that the server knows the user-to-helper cache association, defined as the sets of users connected to each helper cache, during the cache placement phase. We propose a solution for the aforementioned coded caching problem by introducing a combinatorial structure called a Shared and Private Placement Delivery Array (SP-PDA). These SP-PDAs describe the helper cache placement, private cache placement, and the server transmissions in a single array. Further, we propose a novel construction of SP-PDAs using two Placement Delivery Arrays (PDAs). Interestingly, we observe that the permutations of the columns of the two chosen PDAs result in SP-PDAs with different performances. Moreover, we characterize the conditions for selecting the best column permutations of the chosen PDAs. Furthermore, the coded caching schemes resulting from SP-PDAs subsume two existing coded caching schemes as special cases. Additionally, SP-PDAs enable the construction of coded caching schemes with much smaller subpacketization numbers -subpacketization number is defined as the number of subfiles to which a file is divided- compared to the existing schemes, without paying much in terms of rate (the size of the transmission in the delivery phase).

翻译：我们考虑一个编码缓存网络，该网络由一个包含$N$个文件的服务器连接至$K$个用户组成，其中每个用户配备大小为$M_p$单位的专用缓存。此外，该网络包含$\Lambda\leq K$个辅助缓存，每个大小为$M_h$单位。每个辅助缓存可为任意数量的用户提供服务；但每个用户仅能访问单个辅助缓存。同时，我们假设在缓存放置阶段，服务器已知用户-辅助缓存关联关系（即连接到每个辅助缓存的用户集合）。我们通过引入一种名为共享与私有放置分发数组（SP-PDA）的组合结构，为上述编码缓存问题提出解决方案。这些SP-PDA将辅助缓存放置、私有缓存放置及服务器传输统一描述在单一数组中。进一步地，我们提出了一种利用两个放置分发数组（PDA）构建SP-PDA的新方法。有趣的是，我们观察到所选两个PDA的列排列会导致不同性能的SP-PDA。此外，我们刻画了选择最佳列排列的条件。由SP-PDA生成的编码缓存方案将两种现有编码缓存方案作为特例。同时，与现有方案相比，SP-PDA能在速率（即交付阶段的传输大小）几乎不变的情况下，构建出具有更小子包化数（即文件被划分的子文件数量）的编码缓存方案。