  In 2010, Silva, Kschischang and K\"otter studied certain classes of finite
field matrix channels in order to model random linear network coding where
exactly $t$ random errors are introduced.
  In this paper we consider a generalisation of these matrix channels where the
number of errors is not required to be constant, indeed the number of errors
may follow any distribution. We show that a capacity-achieving input
distribution can always be taken to have a very restricted form (the
distribution should be uniform given the rank of the input matrix). This result
complements, and is inspired by, a paper of Nobrega, Silva and Uchoa-Filho,
that establishes a similar result for a class of matrix channels that model
network coding with link erasures. Our result shows that the capacity of our
channels can be expressed as a maximisation over probability distributions on
the set of possible ranks of input matrices: a set of linear rather than
exponential size.
