  A linear Fractional Network Coding (FNC) solution over $\mathbb{F}_q$ is a
linear network coding solution over $\mathbb{F}_q$ in which the message
dimensions need not necessarily be the same and need not be the same as the
edge vector dimension. Scalar linear network coding, vector linear network
coding are special cases of linear FNC. In this paper, we establish the
connection between the existence of a linear FNC solution for a network over
$\mathbb{F}_q$ and the representability over $\mathbb{F}_q$ of discrete
polymatroids, which are the multi-set analogue of matroids. All previously
known results on the connection between the scalar and vector linear
solvability of networks and representations of matroids and discrete
polymatroids follow as special cases. An algorithm is provided to construct
networks which admit FNC solution over $\mathbb{F}_q,$ from discrete
polymatroids representable over $\mathbb{F}_q.$ Example networks constructed
from discrete polymatroids using the algorithm are provided, which do not admit
any scalar and vector solution, and for which FNC solutions with the message
dimensions being different provide a larger throughput than FNC solutions with
the message dimensions being equal.
