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It is known that there exists a multiple-unicast network which has a rate 1 linear network coding solution if and only if the characteristic of the finite field belongs to a given finite or co-finite set of primes. In this paper, we present a generalization of this result for a linear network coding solution of any rate. Specifically, we show that for any non-zero positive rational number k/n, there...
Sum-networks are networks where all the terminals demand the sum of the symbols generated at the sources. It has been shown that for any finite set/co-finite set of prime numbers, there exists a sum-network which has a rate 1 linear network coding solution if and only if the characteristic of the finite field belongs to the given set. It has also been shown that for any positive rational number k/n,...
It is known that the Fano network has a vector linear solution if and only if the characteristic of the finite field is 2; and the non-Fano network has a vector linear solution if and only if the characteristic of the finite field is not 2. Using these properties of Fano and non-Fano networks it has been shown that linear network coding is insufficient. In this paper we generalize the properties of...
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...
Rai et al. in [1] showed that any rational number p/q is the capacity of some sum-network. The authors proved this result by constructing such sum-networks. The authors first constructed a sum-network with capacity 1/q, and then they connected p of such networks in parallel to create a sum-network of capacity p/q. Such construction requires 2q2 - q number of sources and 2q2-q number of terminals....
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