Network Coding Bounds 17.1 Network Coding 17.1.1 Upper Bound

Start with a directed acyclic graph (DAG) with a single sender and many receivers, where each receiver has k-edge connectivity from the sender. We assume that each edge has unit capacity, say, 1 bit/second. If our initial DAG does not meet this criterion, we can easily construct one that does: first, divide edge capacities by their greatest common divisor, then replace each c-capacity edge with c unit-capacity edges. Our unit capacities are no longer in the same units as before, but this is not a problem: the relative capacities have been maintained. Under these conditions, each recipient can receive k bits/second from the sender. Today, we prove two bounds on q, the field size of the bits sent. For ordinary, Boolean bits, q = 2, but we may need a larger field size to handle conflicts among the paths shared by different receivers. First, we will prove the upper bound q = O( # receivers). Second, we will demonstrate the lower bound q = Ω( √ # receivers ).