Projective Geometry based Coded Caching Schemes with Subexponential and Linear Subpacketizations

Abstract

Coded Caching is a recent technique that optimizes the use of a multi-client broadcast channel by the use of local storage available at the clients and by using coded transmissions to serve multiple clients at once. While large gains in the rate of communication are obtained using coded caching, most existing schemes require that the files at the server be divisible into a large number of parts. In particular, most known coded caching schemes require subpacketization $F = {e^{O\left({{K^{\frac{1}{r}}}}\right)}}$, where K is the number of clients and r is some constant positive integer. While few schemes having subpacketization linear in K are known in literature, unfortunately such schemes require large number of users to exist or offer little gain in rate. In this work, we propose a class of coded caching schemes based on projective geometries over finite fields, generalizing recent results. Our construction achieves subexponential (in K) subpacketization, i.e., $F = {q^{O({{({{\log }_q}K)}^2})}}$ , and gain O((log q K)n+1), for large K and the cached fraction $\frac{M}{N}$ being upper bounded by a constant $\frac{{n + 1}}{{{q^{\alpha - n}}}}$ (where α, n being positive integer constants such that n < α and q is some prime power). For specific values of the scheme parameters, we get a new linear subpacketization scheme with the number of clients $K \leq {q^{2{\lambda ^2}{q^2}}}$ (and subpacketization F ), cache fraction $\frac{M}{N} \leq \lambda $, and coded caching gain $\gamma \geq \frac{{{4^{\lambda q}}}}{{{\lambda _q}}}$ where q is some prime power, and λ ∈ (0, 1).