Optimal Placement Delivery Arrays from $t$-Designs with Application to Hierarchical Coded Caching

TL;DR

This work tackles the subpacketization bottleneck in coded caching by introducing two PDA constructions derived from combinatorial $t$-designs, yielding lower transmission rates $R=S/F$ for fixed $K$, $F$, and cache fraction $M/N=Z/F$. Scheme I and Scheme II produce new PDAs with explicit parameters that, in several cases, meet or beat existing $t$-design-based schemes and achieve optimality with respect to established lower bounds on $S$ (RPDA). Building on these PDAs, the authors construct hierarchical PDAs (HPDAs) to enable low-subpacketization two-layer caching, providing explicit rate and memory tradeoffs $R_1$, $R_2$, and $(M_1/N,M_2/N)$ for given design parameters. The results demonstrate practical reductions in subpacketization while maintaining competitive rates, with concrete examples and proofs of optimality in key parameter regimes, illustrating potential for scalable, hierarchical caching implementations. Overall, the paper advances the design of cache networks by connecting combinatorial designs to optimal PDA-based coded caching and its hierarchical extension, enabling more feasible deployments.

Abstract

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.

Figures (5)

• Figure 1: $8\times84$ array obtained from $3-(8,4,1)$ design for $i=2$
• Figure 2: $(84,8,6,24)$ PDA obtained from $3-(8,4,1)$ design for $i=2$
• Figure 3: $(21,7,5,7)$ PDA
• Figure 4: Example of HPDA in Theorem \ref{['thm:HPDA']}
• Figure 5: Example of HPDA in Theorem \ref{['thm:HPDA2']}

Theorems & Definitions (32)

• Definition 1
• Definition 2
• Theorem 1
• Remark 1
• Definition 3: Optimal PDA
• Theorem 2
• Theorem 3
• Example 1
• Example 2
• Example 3