The Capacity of a Finite Field Matrix Channel

Abstract

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.