Hierarchical Coded Caching with Low Subpacketization and Coding Delay using Combinatorial t-Designs

TL;DR

This work targets the subpacketization bottleneck in hierarchical coded caching by introducing a PDAs-based HPDA framework built from combinatorial $t$-designs. By deriving PDAs with subpacketization $F=inom{v}{i}$ and constructing HPDAs with fixed numbers of mirrors and users, it achieves substantial reductions in subpacketization while preserving favorable delivery performance; memory sharing further extends achievable memory-load points. The proposed HPDAs subsume many known PDA constructions and provide improved loads in several regimes, with a clear tradeoff between subpacketization and coding delay depending on concurrency assumptions. Overall, the approach enables practical, scalable hierarchical caching for large networks and IoT scenarios, offering a versatile toolkit for low-subpacketization designs and memory-sharing strategies.

Abstract

Coded caching scheme originally proposed by Maddah-Ali and Niesen (MN) considered a broadcast network consisting of a single server connected to a set of users each having a cache memory. Motivated by practical scenarios, Karamchandani \textit{et al.} in [16] proposed a coded caching scheme for a two-layer hierarchical network consisting of a single server connected to multiple mirror sites and each mirror site connected to a distinct set of users, in which both mirror sites and users having cache memories. Low subpacketization level coded caching schemes are desirable for practical implementations. Placement delivery array (PDA) was proposed as a tool to design coded caching schemes with reduced subpacketization level by Yan \textit{et al.} in [4]. Schemes with reduced subpacketization levels are studied extensively in the literature for single-layer networks. Kong \textit{et al.} in [17] proposed a structure called hierarchical placement delivery arrays (HPDA), which characterizes a hierarchical coded caching system and also proposed a class of HPDAs that gives low subpacketization level schemes by using two PDAs. Low subpacketization level hierarchical schemes using combinatorial $t$-designs is proposed in [20]. Apart from that there is no other existing work that discusses the subpacketization problem in a hierarchical network. This paper proposes a class of HPDA construction that gives low subpacketization level hierarchical coded caching schemes, by first constructing a new class of PDAs. Compared with the existing schemes, in cases where the system parameters and subpacketization level are the same, the proposed hierarchical scheme has a better coding delay. Further, the new class of PDAs constructed either subsumes several known PDA constructions or achieves better transmission load for the same system parameters.

Figures (8)

• Figure 1: The two-layer $(K_1,K_2;M_1,M_2;N)$ hierarchical caching system.
• Figure 2: $(21,7,5,7)$ PDA
• Figure 3: Example of HPDA in Theorem \ref{['thm:HPDA']}
• Figure 4: $15\times 30$ array obtained by Corollary \ref{['cor:HPDA']} for $n=6, k=5, j=4$ and $i=2$
• Figure 5: HPDA obtained for $n=6, k=5, j=4$ and $i=2$ and $(5,5,2,7)$ PDA in Corollary \ref{['cor:hpda']}.

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Lemma 1
• Definition 3
• Lemma 2
• Example 1
• Definition 4
• Definition 5: Design $(\mathcal{X}, \mathcal{A})$
• Definition 6: $t-(v, k, \lambda)$ design
• Example 2