In the $(L,K,M,N)$ cache-aided multiple-input single-output (MISO) broadcast channel (BC) system, the server is equipped with $L$ antennas and communicates with $K$ single-antenna users through a wireless broadcast channel where the server has a library containing $N$ files, and each user is equipped with a cache of size $M$ files. Under the constraints of uncoded placement and one-shot linear delivery strategies, many schemes achieve the maximum sum Degree-of-Freedom (sum-DoF). However, for general parameters $L$, $M$, and $N$, their subpacketizations increase exponentially with the number of users. We aim to design a MISO coded caching scheme that achieves a large sum-DoF with low subpacketization $F$. An interesting combinatorial structure, called the multiple-antenna placement delivery array (MAPDA), can be used to generate MISO coded caching schemes under these two strategies; moreover, all existing schemes with these strategies can be represented by the corresponding MAPDAs. In this paper, we study the case with $F=K$ (i.e., $F$ grows linearly with $K$) by investigating MAPDAs. Specifically, based on the framework of Latin squares, we transform the design of MAPDA with $F=K$ into the construction of a combinatorial structure called the $L$-half-sum disjoint packing (HSDP). It is worth noting that a $1$-HSDP is exactly the concept of NHSDP, which is used to generate the shared-link coded caching scheme with $F=K$. By constructing $L$-HSDPs, we obtain a class of new schemes with $F=K$. Finally, theoretical and numerical analyses show that our $L$-HSDP schemes significantly reduce subpacketization compared to existing schemes with exponential subpacketization, while only slightly sacrificing sum-DoF, and achieve both a higher sum-DoF and lower subpacketization than the existing schemes with linear subpacketization.

翻译：在$(L,K,M,N)$缓存辅助多输入单输出(MISO)广播信道(BC)系统中，服务器配备$L$根天线，通过无线广播信道与$K$个单天线用户通信，服务器拥有包含$N$个文件的库，每个用户配备容量为$M$个文件的缓存。在未编码放置和单次线性传输策略的约束下，许多方案实现了最大和自由度(sum-DoF)。然而，对于一般参数$L$、$M$和$N$，其分组化程度随用户数量呈指数增长。本文旨在设计一种能实现高和自由度且具有低分组化$F$的MISO编码缓存方案。一种称为多天线放置传输阵列(MAPDA)的趣味组合结构可用于在这两种策略下生成MISO编码缓存方案；此外，所有采用这些策略的现有方案均可由对应的MAPDA表示。本文通过研究MAPDA来探讨$F=K$（即$F$随$K$线性增长）的情形。具体而言，基于拉丁方的框架，我们将$F=K$的MAPDA设计转化为一种称为$L$-半和不相交填充(HSDP)的组合结构构造问题。值得注意的是，$1$-HSDP正是NHSDP概念，该概念曾用于生成$F=K$的共享链路编码缓存方案。通过构造$L$-HSDP，我们获得了一类具有$F=K$的新方案。最终，理论与数值分析表明：相较于具有指数级分组化的现有方案，我们的$L$-HSDP方案在仅轻微牺牲和自由度的同时显著降低了分组化程度，并且相比现有线性分组化方案实现了更高的和自由度与更低的分组化水平。