  This paper provides a new duality between entropy functions and network
codes. Given a function $g\geq 0$ defined on all proper subsets of $N$ random
variables, we provide a construction for a network multicast problem which is
solvable if and only if $g$ is entropic. The underlying network topology is
fixed and the multicast problem depends on $g$ only through edge capacities and
source rates. Relaxing the requirement that the domain of $g$ be subsets of
random variables, we obtain a similar duality between polymatroids and the
linear programming bound. These duality results provide an alternative proof of
the insufficiency of linear (and abelian) network codes, and demonstrate the
utility of non-Shannon inequalities to tighten outer bounds on network coding
capacity regions.
