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Capacity Analysis of Linear Operator Channels Over Finite Fields

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4 Author(s)
Shenghao Yang ; Inst. for Theor. Comput. Sci., Tsinghua Univ., Beijing, China ; Siu-Wai Ho ; Jin Meng ; En-hui Yang

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 CSS are analyzed. By establishing and comparing lower bounds on C and upper bounds on CSS, various necessary conditions and sufficient conditions such that C = CSS are obtained. A new class of LOCs such that C = CSS is identified, which includes LOCs with uniform-given-rank transfer matrices as special cases. It is also demonstrated that CSS 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, CSS 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.

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Information Theory, IEEE Transactions on  (Volume:60 ,  Issue: 8 )