Linear Network Coding, Linear Index Coding and Representable Discrete Polymatroids
Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections among linear network coding, linear index coding, and representable discrete polymatroids. We consider the vector linear solutions of networks over a field $\mathbb {F}_{q}$ , with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. It is well known that a scalar linear solution over $\mathbb {F}_{q}$ exists for a network if and only if the network is matroidal with respect to a matroid representable over $\mathbb {F}_{q}$ . We define a discrete polymatroidal network and show that a linear fractional solution over a field $\mathbb {F}_{q}$ exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over $\mathbb {F}_{q}$ . An algorithm to construct the networks starting from certain class of discrete polymatroids is provided. Every representation over $\mathbb {F}_{q}$ for the discrete polymatroid, results in a linear fractional solution over $\mathbb {F}_{q}$ for the constructed network. Next, we consider the index coding problem, which involves a sender, which generates a set of messages $X=\{x_{1},x_{2},\dotso x_{k}\}$ , and a set of receivers $\mathcal {R}$ , which demand messages. A receiver $R \in \mathcal {R}$ is specified by the tuple $(x,H)$ , where $x \in X$ is the message demanded by $R$ and $H \subseteq X \!\setminus \! \{x\}$ is the side information possessed by $R$ . We first show that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid, satisfying certain conditions, which are determined by the index coding problem considered. Rouayheb et al. showed that the problem of finding a multi-linear representation for a matroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that matroid. The multi-linear representation of a matroid can be viewed as a special case of representation of an appropriate discrete polymatroid. We generalize the result of Rouayheb et al. , by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid.