vectorlinear solvability of non-multicast networks depends upon both the characteristic of the finite field and the dimension of the vectorlinearnetwork code. In the literature, the dependency on the characteristic...
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vectorlinear solvability of non-multicast networks depends upon both the characteristic of the finite field and the dimension of the vectorlinearnetwork code. In the literature, the dependency on the characteristic of the finite field and the dependency on the dimension have been studied separately. In this paper, we show the interdependency between the characteristic of the finite field and the dimension of the vectorlinearnetwork code that achieves a vector linear network coding (VLNC) solution in non-multicast networks. For any given network N, we define P(N, d) as the set of all characteristics of finite fields over which the network N has a d-dimensional VLNC solution. To the best of our knowledge, for any network N shown in the literature, if P(N, 1) is non-empty, then P(N, 1) = P(N, d) for any positive integer d. We show that, for any two non-empty sets of primes P-1 and P-2, there exists a network N such that P(N, 1) = P1, but P(N, 2) = {P1, P2}. We also show that there are networks exhibiting a similar advantage (the existence of a VLNC solution over a larger set of characteristics) if the dimension is increased from 2 to 3. However, such behaviour is not universal, as there exist networks which admit a VLNC solution over a smaller set of characteristics of finite fields when the dimension is increased. Using the networks constructed in this paper, we further demonstrate that: (i) a network having an m(1)-dimensional VLNC solution over a finite field of some characteristic and an m(2)-dimensional VLNC solution over a finite field of some other characteristic may not have an (m(1) + m(2))-dimensional VLNC solution over any finite field;(ii) there exist a class of networks for which scalar linearnetworkcoding (SLNC) over non-commutative rings has some advantage over SLNC over finite fields: the least sized non-commutative ring over which each network in the class has an SLNC solution is significantly lesser in size than the least sized finite field ove
It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vectorlinear solution when message dimension is 2. It is not known whether this result can be gener...
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It is known that there exists a network which does not have a scalar linear solution over any finite field but has a vectorlinear 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 vectorlinear solution, where m is a positive integer greater than or equal to 2, but does not have a vectorlinear solution over any finite field when the message dimension is less than m.
Determining the capacity of multi-receiver networks with arbitrary message demands is an open problem in the networkcoding literature. In this paper, we consider a multi-source, multi-receiver symmetric deterministic...
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ISBN:
(纸本)9781467302234;9781467302241
Determining the capacity of multi-receiver networks with arbitrary message demands is an open problem in the networkcoding literature. In this paper, we consider a multi-source, multi-receiver symmetric deterministic network model parameterized by channel coefficients (inspired by wireless network flow) in which the receivers compute a sum of the symbols generated at the sources. Scalar and vectorlinearcoding strategies are analyzed. It is shown that computation alignment over finite field vector spaces is necessary to achieve the computation capacities in the network. To aid in the construction of coding strategies, network equivalence theorems are established for the decomposition of deterministic models into elementary sub-networks. The linearcoding capacity for computation is characterized for all channel parameters considered in the model for a countably infinite class of networks. The constructive coding schemes introduced herein for a specific class of networks provide an optimistic viewpoint for the application of structured codes in network communication.
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