The Additive-Multiplicative Matrix Channel (AMMC) was introduced by Silva, Kschischang and Kötter in 2010 to model data transmission using random linear network coding. The input and output of the channel are$n\times m$matrices over a finite field$\mathbb {F}_{q}$. When the matrix X is input, the channel outputs$Y=A(X+W)$where A is a uniformly chosen$n\times n$invertible matrix over$\mathbb {F}_{q}$and where W is a uniformly chosen$n\times m$matrix over$\mathbb {F}_{q}$of rank t. Silva et al. considered the case when$2n\leq m$. They determined the asymptotic capacity of the AMMC when t, n and m are fixed and$q\rightarrow \infty $. They also determined the leading term of the capacity when q is fixed, and t, n and m grow linearly. We generalise these results, showing that the condition$2n\geq m$can be removed. (Our formula for the capacity falls into two cases, one of which generalises the$2n\geq m$case.) We also improve the error term in the case when q is fixed.

In 2010, Silva et al. 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 generalization 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 et al., which 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 maximization over probability distributions on the set of possible ranks of input matrices: a set of linear rather than exponential size.