Coded Caching for Two-Dimensional Multi-Access Networks With Cyclic Wrap Around

This paper studies a novel multi-access coded caching (MACC) model in the two-dimensional (2D) topology, which is a generalization of the one-dimensional (1D) MACC model proposed by Hachem et al. The 2D MACC model is formed by a server containing <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> files, <inline-formula> <tex-math notation="LaTeX">$K_{1}\times K_{2}$ </tex-math></inline-formula> cache-nodes with <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> files located at a grid with <inline-formula> <tex-math notation="LaTeX">$K_{1}$ </tex-math></inline-formula> rows and <inline-formula> <tex-math notation="LaTeX">$K_{2}$ </tex-math></inline-formula> columns, and <inline-formula> <tex-math notation="LaTeX">$K_{1}\times K_{2}$ </tex-math></inline-formula> cache-less users where each user is connected to <inline-formula> <tex-math notation="LaTeX">$L^{2}$ </tex-math></inline-formula> nearby cache-nodes. The server is connected to the users through an error-free shared link, while the users can retrieve the cached content of the connected cache-nodes without cost. Our objective is to minimize the worst-case transmission load over all possible users’ demands. In this paper, we first propose a grouping scheme for the case where <inline-formula> <tex-math notation="LaTeX">$K_{1}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$K_{2}$ </tex-math></inline-formula> are divisible by <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>. By partitioning the cache-nodes and users into <inline-formula> <tex-math notation="LaTeX">$L^{2}$ </tex-math></inline-formula> groups such that no two users in the same group share any cache-node, we use the shared-link coded caching scheme proposed by Maddah-Ali and Niesen for each group. Then for any model parameters satisfying <inline-formula> <tex-math notation="LaTeX">$\min \{K_{1},K_{2}\}\geq L$ </tex-math></inline-formula>, we propose a transformation approach which constructs a 2D MACC scheme from two classes of 1D MACC schemes in vertical and horizontal projections, respectively. As a result, we can construct 2D MACC schemes that achieve maximum local caching gain and improved coded caching gain, compared to the baseline scheme by a direct extension from 1D MACC schemes. In addition, we propose new information theoretic converse bounds under the uncoded placement constraint by leveraging the network topology.