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Path Gain Algebraic Formulation for the scalar linear network coding Problem

IEEE TRANSACTIONS ON INFORMATION THEORY 2010年第9期56卷 4520-4531页

作者： Subramanian, Abhay T. Thangaraj, Andrew Stanford Univ Dept Management Sci & Engn Stanford CA 94305 USA Indian Inst Technol Dept Elect Engn Chennai 600036 Tamil Nadu India

In the algebraic view, the solution to a network coding problem is seen as a variety specified by a system of polynomial equations typically derived by using edge-to-edge gains as variables. The output from each sink is equated to its demand to obtain polynomial equations. In this paper, we propose a method to derive the polynomial equations using source-to-sink path gains as the variables. In the path gain formulation, we show that linear and quadratic equations suffice;therefore, network coding becomes equivalent to a system of polynomial equations of maximum degree 2. We present algorithms for generating the equations in the path gains and for converting path gain solutions to edge-to-edge gain solutions. Because of the low degree, simplification is readily possible for the system of equations obtained using path gains. Using small-sized network coding problems, we show that the path gain approach results in simpler equations and determines solvability of the problem in certain cases. On a larger network (with 87 nodes and 161 edges), we show how the path gain approach continues to provide deterministic solutions to some network coding problems.

关键词： Algebraic network coding network coding scalar linear network coding

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On the Message Dimensions of Vector linearly Solvable networks

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IEEE COMMUNICATIONS LETTERS 2016年第9期20卷 1701-1704页

作者： Das, Niladri Rai, Brijesh Kumar IIT Guwahati Dept Elect & Elect Engn Gauhati 781039 India

It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vector linear solution when message dimension is 2. It is not known whether this result can be generalized for an arbitrary message dimension. In this letter, we show that there exists a network that admits an m dimensional vector linear solution, where m is a positive integer greater than or equal to 2, but does not have a vector linear solution over any finite field when the message dimension is less than m.

关键词： scalar linear network coding vector linear network coding Matroid theory discrete polymatroids

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