  We consider communication over a noisy network under randomized linear
network coding. Possible error mechanism include node- or link- failures,
Byzantine behavior of nodes, or an over-estimate of the network min-cut.
Building on the work of Koetter and Kschischang, we introduce a probabilistic
model for errors. We compute the capacity of this channel and we define an
error-correction scheme based on random sparse graphs and a low-complexity
decoding algorithm. By optimizing over the code degree profile, we show that
this construction achieves the channel capacity in complexity which is jointly
quadratic in the number of coded information bits and sublogarithmic in the
error probability.
