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Caching prefetches some library content at users' memories during theoff-peak times (i.e., {\it placement phase}), such that the number oftransmissions during the peak-traffic times (i.e., {\it delivery phase}) arereduced. A coded caching strategy was originally proposed by Maddah-Ali andNiesen (MN) leading to a multicasting gain compared to the conventional uncodedcaching, where each message in the delivery phase is useful to multiple userssimultaneously. However, the MN coded caching scheme suffers from the highsubpacketization which makes it impractical. In order to reduce thesubpacketization while retain the multicast opportunities in the deliveryphase, Yan et al. proposed a combinatorial structure called placement deliveryarray (PDA) to design coded caching schemes. In this paper, we propose twonovel frameworks for constructing PDA via Cartesian product, which constructs aPDA 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 novelcaching schemes are obtained which can significantly reduce thesubpacketization of the MN scheme while slightly increasing the needed numberof transmissions. For instance, for the third scheme which works for any numberof 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.

Novel Frameworks for Coded Caching via Cartesian Product with Reduced Subpacketization