Caching prefetches some library content at users' memories during the off-peak times (i.e., {\it placement phase}), such that the number of transmissions during the peak-traffic times (i.e., {\it delivery phase}) are reduced. A coded caching strategy was originally proposed by Maddah-Ali and Niesen (MN) leading to a multicasting gain compared to the conventional uncoded caching, where each message in the delivery phase is useful to multiple users simultaneously. However, the MN coded caching scheme suffers from the high subpacketization which makes it impractical. In order to reduce the subpacketization while retain the multicast opportunities in the delivery phase, Yan et al. proposed a combinatorial structure called placement delivery array (PDA) to design coded caching schemes. In this paper, we propose two novel frameworks for constructing PDA via Cartesian product, which constructs a PDA for $mK_1$ users by the $m$-fold Cartesian product of a PDA for $K_1$ users. By applying the proposed frameworks to some existing PDAs, three novel caching schemes are obtained which can significantly reduce the subpacketization of the MN scheme while slightly increasing the needed number of transmissions. For instance, for the third scheme which works for any number of users and any memory regime, while reducing the coded caching gain by one, the needed subpacketization is at most $O\left(\sqrt{\frac{K}{q}}2^{-\frac{K}{q}}\right)$ of that of the MN scheme, where $K$ is the number of users, $0<z/q<1$ is the memory ratio of each user, and $q,z$ are coprime.

翻译：Maddah-Ali 和 Niesen (MN) 最初提出了一个代码化缓冲策略, 与传统的未编码缓冲相比, 交付阶段的每条信息都同时对多个用户有用。 然而, MN 编码缓冲计划受到高分包化的影响, 这使得它变得不切实际。 为了减少次包装化,同时保留交付阶段的多版机会, Yan 等人 提议了一个叫做 交付阵列(PDA) 的组合结构, 以设计编码化计划。 在本文中, 我们提出了两个新框架, 用于通过卡通产品建造 PDA, 由美元/一元的卡通产品为 $K_1, 使PDA 的美元分包化进程变得不切实际。 用于美元- 美元- 1美元的用户, 正在将现有的递解框架称为 PDA 。