The Capacity of a Finite Field Matrix Channel
Simon R. Blackburn⋆, Jessica Claridge · IEEE Transactions on Information Theory · 2025
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.