Caching and Coded Multicasting: Multiple Groupcast Index Coding

The capacity of caching networks has received considerable attention in the past few years. A particularly studied setting is the case of a single server (e.g., a base station) and multiple users, each of which caches segments of files in a finite library. Each user requests one (whole) file in the library and the server sends a common coded multicast message to satisfy all users at once. The problem consists of finding the smallest possible codeword length to satisfy such requests. In this paper we consider the generalization to the case where each user places $L \geq 1$ requests. The obvious naive scheme consists of applying $L$ times the order-optimal scheme for a single request, obtaining a linear in $L$ scaling of the multicast codeword length. We propose a new achievable scheme based on multiple groupcast index coding that achieves a significant gain over the naive scheme. Furthermore, through an information theoretic converse we find that the proposed scheme is approximately optimal within a constant factor of (at most) $18$.