Optimality of Linear Codes for Broadcast-Mode Multicast Networks

It is known that linear codes are sufficient to solve the multicast network coding problem when each out-edge of a network node carries its own specific function of the in-edges of the node, i.e. operating in "point-to-point-mode." Alternatively, in "broadcast-mode," a network has the property that for each node, every out-edge of the node carries the same function of the in-edges of the node. Only one transmission is required in order to send the same function on all of the out-edges of a node. The edge functions in broadcast-mode can vary from node to node and each edge can carry an arbitrary number of transmissions, with at most one per time unit. We prove that linear codes are sufficient, in terms of total number of transmissions, for multicast networks in broadcast-mode. That is, we show that for any broadcast-mode solution to a multicast network, there exists a linear broadcast-mode solution over some finite field which does not increase the total number of network transmissions