A New Construction Structure on Coded Caching with Linear Subpacketization: Non-Half-Sum Latin Rectangle

TL;DR

This work introduces Non-Half-Sum Latin Rectangles (NHSLR) as a novel combinatorial structure to extend linear subpacketization coded caching beyond the NHSDP paradigm. By mapping NHSLR to Plac ement Delivery Arrays (PDAs) through a construction that uses a diagonal matrix X, the authors obtain a broad family of $(K,M,N)$ coded caching schemes with subpacketization $F=vg$ and transmission load $R=b/g$, while the memory ratio is $M/N=1-b/v$ and $K=v$. Theoretical results show how to select parameters to maximize multicast gains under linear subpacketization, and a sub-optimal closed-form solution yields practical schemes with $K=q^n$, $F=2^n q^n$, $g=2^n$, and $R=(\frac{\lfloor q-1\rfloor}{2})^n$. Numerical comparisons demonstrate that NHSLR-based schemes outperform existing linear-subpacketization schemes in transmission load and can approach the performance of some exponential-subpacketization schemes for certain parameter ranges, representing a meaningful bridge between low-subpacketization and low-load regimes in coded caching.

Abstract

Coded caching is recognized as an effective method for alleviating network congestion during peak periods by leveraging local caching and coded multicasting gains. The key challenge in designing coded caching schemes lies in simultaneously achieving low subpacketization and low transmission load. Most existing schemes require exponential or polynomial subpacketization levels, while some linear subpacketization schemes often result in excessive transmission load. Recently, Cheng et al. proposed a construction framework for linear coded caching schemes called Non-Half-Sum Disjoint Packing (NHSDP), where the subpacketization equals the number of users $K$. This paper introduces a novel combinatorial structure, termed the Non-Half-Sum Latin Rectangle (NHSLR), which extends the framework of linear coded caching schemes from $F=K$ (i.e., the construction via NHSDP) to a broader scenario with $F=\mathcal{O}(K)$. By constructing NHSLR, we have obtained a new class of coded caching schemes that achieves linearly scalable subpacketization, while further reducing the transmission load compared with the NHSDP scheme. Theoretical and numerical analyses demonstrate that the proposed schemes not only achieves lower transmission load than existing linear subpacketization schemes but also approaches the performance of certain exponential subpacketization schemes.

Figures (8)

• Figure 1: $(K,M,N)$ caching system
• Figure 2: Flow diagram for constructing a PDA via $(7,3,4)$ NHSLR
• Figure 4: Memory ratio-subpacketization tradeoff for $K=75$
• Figure 5: Memory ratio-load tradeoff for $K=75$
• Figure 6: Memory ratio-subpacketization tradeoff for $K=343$

Theorems & Definitions (16)

• Definition 1: PDA,YCTC
• Lemma 1: Scheme via PDA,YCTC
• Definition 2: Non-half-sum disjoint packing, NHSDP CWWC
• Lemma 2: CWWC
• Definition 3: Non-half-sum latin rectangle, NHSLR
• Example 1
• Theorem 1: PDA via NHSLR
• Remark 1
• Theorem 2: NHSLR via NHSDP
• Remark 2