The conditions to determine convolutional network coding on matrix representation

Over acyclic networks, it is well known that the global encoding kernels are uniquely determined by the local encoding kernels. But it is not in the case over cyclic networks. To study this problem, we employ matrix power series to describe the encoding kernels. This arrangement not only makes the physical meaning explicitly, but also makes it easy to obtain the conditions of determining the global encoding kernels from the local encoding kernels. We denote by K0 the constant term of the local encoding kernel matrix. Then the above conditions are characteristic of K0. It is shown that a nilpotent K0 is sufficient to determine F(z). K0 is nilpotent when the encoding topology with respect to K0 is acyclic. This result facilitates convolutional network coding encoder design. Then the equivalent conditions to determine convolutional network coding are deduced, and the inclusion relations among these conditions are further discussed in some examples.