Valuable messages and random outputs of channels in linear network coding

Motivated by the fact that the most problems on network coding can be represented as how much information about a given subset of network inputs can be obtained by legal or illegal users from the channels accessed by them, in this paper we investigate the relation between a subset of random network inputs and the outputs of an arbitrarily given set of channels in networks. We focus on linear network codes because they are widely studied and applied. We begin with the algebraic structure of cosets of linear subspaces and derive bounds on their mutual information and the conditions for their tightness. To apply the results to random linear network coding we introduce strongly generic linear network codes such that for sufficiently large coding fields a random linear network code is strongly generic with high probability. Our results show that random linear network coding is good for error correction and security but not efficient for multiple source network coding.