  Motivated by communication through a network employing linear network coding,
capacities of linear operator channels (LOCs) with arbitrarily distributed
transfer matrices over finite fields are studied. Both the Shannon capacity $C$
and the subspace coding capacity $C_{\text{SS}}$ are analyzed. By establishing
and comparing lower bounds on $C$ and upper bounds on $C_{\text{SS}}$, various
necessary conditions and sufficient conditions such that $C=C_{\text{SS}}$ are
obtained. A new class of LOCs such that $C=C_{\text{SS}}$ is identified, which
includes LOCs with uniform-given-rank transfer matrices as special cases. It is
also demonstrated that $C_{\text{SS}}$ is strictly less than $C$ for a broad
class of LOCs. In general, an optimal subspace coding scheme is difficult to
find because it requires to solve the maximization of a non-concave function.
However, for a LOC with a unique subspace degradation, $C_{\text{SS}}$ can be
obtained by solving a convex optimization problem over rank distribution.
Classes of LOCs with a unique subspace degradation are characterized. Since
LOCs with uniform-given-rank transfer matrices have unique subspace
degradations, some existing results on LOCs with uniform-given-rank transfer
matrices are explained from a more general way.
