  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 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. We
define a \textit{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
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 and 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. El Rouayheb et. al. showed
that the problem of finding a multi-linear representation for a matroid can be
reduced to finding a \textit{perfect linear index coding solution} for an index
coding problem obtained from that matroid. We generalize the result of El
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.
