Network Coding from a Network Flow Perspective

The algebraic framework introduced in [4] gives an algebraic condition for the feasibility of a set of multicast connections in a network, that is equivalent to the max-flow min-cut condition of [1]. The algebraic condition also checks the validity of a linear coding solution to a given multicast connection problem. We present two alternative formulations of this condition that are closely tied to network flow, providing further insights and connections with combinatorial optimization. They are also easier to work with in many cases as they do not involve matrix products and inversions. From these results we derive a substantially tighter upper bound on the coding field size required for a given connection problem than that in [4].