Capacity Analysis of Linear Operator Channels Over Finite Fields

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_SS are analyzed. By establishing and comparing lower bounds on C and upper bounds on C_SS, various necessary conditions and sufficient conditions such that C = C_SS are obtained. A new class of LOCs such that C = C_SS is identified, which includes LOCs with uniform-given-rank transfer matrices as special cases. It is also demonstrated that C_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 nonconcave function. However, for an LOC with a unique subspace degradation, C_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.