Abstract-A new inner bound on the capacity region of the general index coding problem is established. Unlike most existing bounds that are based on graph theoretic or algebraic tools, the bound relies on a random coding scheme and optimal decoding, and has a simple polymatroidal single-letter expression. The utility of the inner bound is demonstrated by examples that include the capacity region for all index coding problems with up to five messages (there are 9846 nonisomorphic ones).

In this paper, we study the capacity region of the general distributed index coding. In contrast to the traditional centralized index coding where a single server contains all n messages requested by the receivers, in the distributed index coding there are 2 n − 1 servers, each containing a unique non-empty subset J of the messages and each is connected to all receivers via a noiseless independent broadcast link with an arbitrary capacity CJ ≥ 0. First, we generalize the existing polymatroidal outer bound on the capacity region of the centralized problem to the distributed case. Next, building upon the existing centralized composite coding scheme, we propose three distributed composite coding schemes and derive the corresponding inner bounds on the capacity region. We present a number of interesting numerical examples, which highlight the subtleties and challenges of dealing with the distributed index coding, even for very small problem sizes of n = 3 and n = 4.

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The capacity region of the index coding problem is characterized through the notion of confusion graph and its fractional chromatic number. Based on this multiletter characterization, several structural properties of the capacity region are established, some of which are already noted by Tahmasbi, Shahrasbi, and Gohari, but proved here with simple and more direct graph-theoretic arguments. In particular, the capacity region of a given index coding problem is shown to be simple functionals of the capacity regions of smaller subproblems when the interaction between the subproblems is none, one-way, or complete.

The distributed index coding problem is studied, whereby multiple messages are stored at different servers to be broadcast to receivers with side information. First, the existing composite coding scheme is enhanced for the centralized (single-server) index coding problem, which is then merged with fractional partitioning of servers to yield a new coding scheme for distributed index coding. New outer bounds on the capacity region are also established. For 213 out of 218 non-isomorphic distributed index coding problems with four messages the achievable sum-rate of the proposed distributed composite coding scheme matches the outer bound, thus establishing the sum-capacity for these problems.

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Three science and engineering problems of recent interests-index coding, locally recoverable distributed storage, and guessing games on graphs-are discussed and the connection between their optimal solutions is elucidated. By generalizing recent results by Shanmugam and Dimakis and by Mazumdar on the complementarity between the optimal broadcast rate of an index coding problem on a directed graph and the normalized rate of a locally recoverable distributed storage problem on the same graph, it is shown that the capacity region and the optimal rate region of these two problems are complementary. The main ingredients in establishing this result are the notion of confusion graph introduced by Alon et al. (2008), the vertex transitivity of a confusion graph, the characterization of the index coding capacity region via the fractional chromatic number of confusion graphs, and the characterization of the optimal rate region of the locally recoverable distributed storage via the independence number of confusion graphs. As the third and final facet of the complementarity, guessing games on graphs by Riis are discussed as special cases of the locally recoverable distributed storage problem, and it is shown that the winning probability of the optimal strategy for a guessing game and the ratio between the winning probabilities of the optimal strategy and a random guess can be characterized, respectively, by the capacity region for index coding and the optimal rate region for distributed storage.