Novel Frameworks for Coded Caching via Cartesian Product with Reduced Subpacketization

TL;DR

This paper introduces two new frameworks for constructing placement delivery arrays (PDAs) using Cartesian products, significantly reducing subpacketization in coded caching schemes while maintaining most of the multicast gain.

Contribution

The paper proposes novel Cartesian product-based frameworks for PDA construction, enabling reduced subpacketization in coded caching schemes with minimal loss in efficiency.

Findings

Reduces subpacketization to at most $O(rac{ ext{sqrt}(K)}{q} 2^{-rac{K}{q}})$ of the MN scheme.

Creates three new caching schemes with significantly lower subpacketization.

Maintains near-optimal multicast gain with slight increase in transmission count.

Abstract

Caching prefetches some library content at users' memories during the off-peak times (i.e., {\it placement phase}), such that the number of transmissions during the peak-traffic times (i.e., {\it delivery phase}) are reduced. A coded caching strategy was originally proposed by Maddah-Ali and Niesen (MN) leading to a multicasting gain compared to the conventional uncoded caching, where each message in the delivery phase is useful to multiple users simultaneously. However, the MN coded caching scheme suffers from the high subpacketization which makes it impractical. In order to reduce the subpacketization while retain the multicast opportunities in the delivery phase, Yan et al. proposed a combinatorial structure called placement delivery array (PDA) to design coded caching schemes. In this paper, we propose two novel frameworks for constructing PDA via Cartesian product, which constructs a PDA for $mK_1$ users by the $m$-fold Cartesian product of a PDA for $K_1$ users. By applying the proposed frameworks to some existing PDAs, three novel caching schemes are obtained which can significantly reduce the subpacketization of the MN scheme while slightly increasing the needed number of transmissions. For instance, for the third scheme which works for any number of users and any memory regime, while reducing the coded caching gain by one, the needed subpacketization is at most $O\left(\sqrt{\frac{K}{q}}2^{-\frac{K}{q}}\right)$ of that of the MN scheme, where $K$ is the number of users, $0<z/q<1$ is the memory ratio of each user, and $q,z$ are coprime.