On the capacity of multiplicative finite-field matrix channels

This paper deals with the multiplicative finite-field matrix channel, a discrete memoryless channel whose input and output are matrices (over a finite field) related by a multiplicative transfer matrix. Our model allows this transfer matrix to have any rank, while assuming that all transfer matrices with the same rank are equiprobable. While in general the capacity cannot be obtained in closed form, we provide a simplification of the problem (from q^nm to O(n) variables) which allows for easy numerical computation. A tight upper bound on the capacity is also derived, and for the special case of constant-rank input, we obtain an exact formula. Several existing results can be obtained as special cases of our approach. In addition, we prove that the well-known approach of treating inputs and outputs as subspaces is information-lossless even in this more general case.