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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...
Consider the following scenario: A data storage service provider provides an erasure code based distributed storage system (DSS). For the same data, the service provider gives several options: an (ni, ki) erasure code based DSS for i = 1,2, …, m. The service provider charges differently for different options (say dollar Pi for an (ni, ki) erasure code based DSS for the data B of size |B|). A client...
Suppose that a data storage service provider implements a functional MSR code based distributed storage system (DSS). Consider a scenario where the service provider has implemented an (n1; k1;B; d1; t1) functional MSR code based DSS, where B is the file that is stored. At a later point of time, due to certain considerations (such as storage cost, download bandwidth to repair failed nodes etc.), the...
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....
In this paper, we consider sum-networks, which is a class of directed acyclic networks where each terminal in the network requires sum of all the sources. In the literature, all reported sum-networks are shown to have only certain rational valued coding capacities. For example, it has been shown that a sum-network, having 3 sources and 3 terminals, has coding capacity either 0, 2 over 3 or ≥ 1. It...
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