The Capacity of a Finite Field Matrix Channel

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 × m matrices over a finite field F_q. When the matrix X is input, the channel outputs Y = A(X + W) where A is a uniformly chosen n × n invertible matrix over F_q and where W is a uniformly chosen n × m matrix over F_q of rank t. Silva et al. considered the case when 2n ≤ m. They determined the asymptotic capacity of the AMMC when t, n and m are fixed and q ⟶ ∞. 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 ≥ m can be removed. (Our formula for the capacity falls into two cases, one of which generalises the 2n ≥ m case.) We also improve the error term in the case when q is fixed.