  Motivated by linear network coding, communication channels perform linear
operation over finite fields, namely linear operator channels (LOCs), are
studied in this paper. 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. There are NO constraints on the distribution of the
transformation matrix and the field size.
  Specifically, the optimality of subspace coding over LOCs is investigated. A
lower bound on the maximum achievable rate of subspace coding is obtained and
it is shown to be tight for some cases. The maximum achievable rate of
constant-dimensional subspace coding is characterized and the loss of rate
incurred by using constant-dimensional subspace coding is insignificant.
  The maximum achievable rate of channel training is close to the lower bound
on the maximum achievable rate of subspace coding. Two coding approaches based
on channel training are proposed and their performances are evaluated. Our
first approach makes use of rank-metric codes and its optimality depends on the
existence of maximum rank distance codes. Our second approach applies linear
coding and it can achieve the maximum achievable rate of channel training. Our
code designs require only the knowledge of the expectation of the rank of the
transformation matrix. The second scheme can also be realized ratelessly
without a priori knowledge of the channel statistics.
