  This paper is motivated by the problem of error control in network coding
when errors are introduced in a random fashion (rather than chosen by an
adversary). An additive-multiplicative matrix channel is considered as a model
for random network coding. The model assumes that n packets of length m are
transmitted over the network, and up to t erroneous packets are randomly chosen
and injected into the network. Upper and lower bounds on capacity are obtained
for any channel parameters, and asymptotic expressions are provided in the
limit of large field or matrix size. A simple coding scheme is presented that
achieves capacity in both limiting cases. The scheme has decoding complexity
O(n^2 m) and a probability of error that decreases exponentially both in the
packet length and in the field size in bits. Extensions of these results for
coherent network coding are also presented.
