Network coding

Abstract

In the network coding problem, there are k commodities each with an associated message Mi, a set of sources that know Mi and a set of sinks that request Mi. Each edge in the graph may transmit any function of the messages. These functions define a network coding solution. We explore three topics related to network coding. First, for a model in which the messages and the symbols transmitted on edges are all from the same alphabet [sigma], we prove lower bounds on [the absolute value of sigma]. In one case, we prove [the absolute value of sigma] needs to be doubly-exponential in the size of the network. We also show that it is NP-hard to determine the smallest alphabet size admitting a solution. We then explore the types of functions that admit solutions. In a linear solution over a finite field F the symbol transmitted over each edge is a linear combination of the messages. We show that determining if there exists a linear solution is NP-hard for many classes of network coding problems. As a corollary, we obtain a solvable instance of the network coding problem that does not admit a linear solution over any field F. We then define a model of network coding in which messages are chosen from one alphabet, [gamma], and edges transmit symbols from another alphabet, [sigma]. In this model, we define the rate of a solution as log [gamma absolute value]/ log [sigma absolute value]. We then explore techniques to upper bound the maximum achievable rate for instances defined on directed and undirected graphs. We present a network coding instance in an undirected graph in which the maximum achievable rate is strictly smaller than the sparsity of the graph.