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 [3]. It is not known whether this result can be generalized for an arbitrary message dimension. In this paper, we show that there exists a network which 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.
We consider directed acyclic networks where each terminal requires sum of all the sources. Such a class of networks has been termed as sum-networks in the literature. A sumnetwork having m sources and n terminals has been termed as a ms/nt sum-network. There has been previous works on the capacity of sum-networks, specifically, it has been shown that the capacity of a 3s/3t sum-network is either 0, 2/3 or ≥ 1. In this paper, we consider some generalizations of 3s/3t sum-networks, namely, ms/3t and 3s/nt sum-networks, where m, n ≥ 3. For ms/3t and 3s/nt sum-networks, where m, n ≥ 3, if the mincut between each source and each terminal is at least 1, the capacity is known to be at least 2/3. In this paper, we show that there exist ms/3t and 3s/nt sum-networks whose capacities lie between 2/3 and 1. Specifically, we show that for any positive integer k ≥ 2, there exists a ms/3t sum-network (and also a 3s/nt sum-network) whose capacity is k k+1 . We conjecture that the capacity of a ms/3t sum-network, where m > 3 (and also of a 3s/nt sum-network, where n > 3) is either 0, ≥ 1 or of the form k k+1 , where k is a positive integer greater than or equal to 2.
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