Coded caching scheme originally proposed by Maddah-Ali and Niesen (MN) achieves an optimal transmission rate $R$ under uncoded placement but requires a subpacketization level $F$ which increases exponentially with the number of users $K$ where the number of files $N \geq K$. Placement delivery array (PDA) was proposed as a tool to design coded caching schemes with reduced subpacketization level by Yan \textit{et al.} in \cite{YCT}. This paper proposes two novel classes of PDA constructions from combinatorial $t$-designs that achieve an improved transmission rate for a given low subpacketization level, cache size and number of users compared to existing coded caching schemes from $t$-designs. A $(K, F, Z, S)$ PDA composed of a specific symbol $\star$ and $S$ non-negative integers corresponds to a coded caching scheme with subpacketization level $F$, $K$ users each caching $Z$ packets and the demands of all the users are met with a rate $R=\frac{S}{F}$. For a given $K$, $F$ and $Z$, a lower bound on $S$ such that a $(K, F, Z, S)$ PDA exists is given by Cheng \textit{et al.} in \cite{MJXQ} and by Wei in \cite{Wei} . Our first class of proposed PDA achieves the lower bound on $S$. The second class of PDA also achieves the lower bound in some cases. From these two classes of PDAs, we then construct hierarchical placement delivery arrays (HPDA), proposed by Kong \textit{et al.} in \cite{KYWM}, which characterizes a hierarchical two-layer coded caching system. These constructions give low subpacketization level schemes.

翻译：编码缓存方案由Maddah-Ali和Niesen（MN）首次提出，在未编码放置条件下可实现最优传输速率$R$，但要求子包化级别$F$随用户数$K$呈指数增长（其中文件数$N \geq K$）。Yan等人于文献YCT中提出放置递送阵列（PDA）作为设计低子包化级别编码缓存方案的工具。本文基于组合$t$-设计提出两类新型PDA构造方法，与现有基于$t$-设计的编码缓存方案相比，在给定低子包化级别、缓存容量和用户数条件下实现了更优的传输速率。一个$(K, F, Z, S)$ PDA包含特定符号$\star$和$S$个非负整数，对应一个子包化级别为$F$、$K个用户各缓存$Z$个数据包、所有用户需求以速率$R=S/F$得到满足的编码缓存方案。Cheng等人在文献MJXQ和Wei在文献Wei中分别给出了存在$(K, F, Z, S)$ PDA时$S$的下界。本文第一类PDA构造达到该$S$下界，第二类PDA在某些情况下同样达到下界。基于这两类PDA，我们进一步构造了Kong等人在文献KYWM中提出的分层放置递送阵列（HPDA），该结构表征了双层分层编码缓存系统。这些构造实现了低子包化级别的方案。