Combinatorial Analysis of Coded Caching Schemes

TL;DR

The paper addresses optimizing coded caching under subpacketization constraints by modeling placement and delivery with placement delivery arrays (PDAs) and their optimal subclass, RPDA. It defines the metric $s(F,K,Z)$ as the minimal symbol set size needed for an $S$-PDA$(F,K,Z)$ and develops combinatorial RPDA constructions to achieve this bound, including infinite families and transposition techniques to generate new PDAs for large $F$. The authors provide exact values and tight bounds for $s(F,K,Z)$ across several regimes (small/large $F$, fixed $t$, and special $(K,Z)$ pairs), along with numerous concrete PDAs and recursive constructions. The results yield practical benchmarks and design guidelines for coded caching schemes constrained by subpacketization, aiding efficient cache-enabled network design.

Abstract

Coded caching schemes are used to reduce computer network traffics in peak time. To determine the efficiency of the schemes, \cite{MN} defined the information rate of the schemes and gave a construction of optimal coded caching schemes. However, their construction needs to split the data into a large number of packets which may cause constraints in real applications. Many researchers then constructed new coded caching schemes to reduce the number of packets but that increased the information rate. We define an optimization of coded caching schemes under the limitation of the number of packets which may be used to verify the efficiency of these schemes. We also give some constructions for several infinite classes of optimal coded caching schemes under the new definition.

Figures (11)

• Figure 1: An RPDA$(6,20,3)$
• Figure 2: TB order of combination
• Figure 3: $s(4,K,2)$ for small $K$
• Figure 4: $s(5,K,3)$ for small $K$
• Figure 5: $s(5,K,2)$ for small $K$

Theorems & Definitions (61)

• Example 1.1
• Definition 2.1
• Example 2.2
• Definition 2.3
• Lemma 2.4
• Example 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• Corollary 2.9