Optimality of subspace coding for linear operator channels over finite fields

Motivated by random linear network coding, we study the communication through channels, called linear operator channels (LOCs), that perform linear operation over finite fields. For such a channel, its output vector is a linear transform of its input vector, and the transformation matrix is randomly and independently generated. The transformation matrix is assumed to remain constant for every T input vectors and to be unknown to both the transmitter and the receiver. We study LOCs with arbitrary distributions of transformation matrices and focus on the optimality of subspace coding. We obtain a lower bound on the maximum achievable rate of subspace coding and prove that the bound is asymptotically tight when T goes to infinity. Moreover, this lower bound is tight for regular LOCs when T is sufficiently large. We further show that the loss of rate by using constant-dimensional subspace coding is marginal for practical channel parameters.