Abstract
This paper introduces a relationship between the independence of polynomials associated with the links of the network, and the Jacobian determinant of these polynomials. Also, it presents a way to simplify a given communication network through an algorithm that splits the network into subnets and reintegrates them into a network that is a general representation or model of the studied network. This model is also represented through a combination of polynomial equations and uses Groebner bases to reach a new simplified network equivalent to the given network, which may make studying the ability to solve the problem of network coding less expensive and much easier.
Highlights
Communication is the exchange of information between individuals by different means of transmission
The concept of network coding emerged as an important field of study in the research presented by Ahlswede, Cai Li, and Yeung, 1
This paper has shown that it is possible to accomplish the network coding if and only if the coding vectors for each channel in the network are linearly independent
Summary
Communication is the exchange of information between individuals by different means of transmission. In (Fig. 1b), the nodes linearly combine their inputs at BD and GH, and the receivers observe linear combinations of the source symbols determined by matrices Ci. The main theorem in network coding 12 Theorem 3: Consider a directed graph without circles with unit-capacity edges, h unit-rate information sources and N receivers, such that there are h edgedisjoint paths from the sources to all receivers. There exists a multicast transmission scheme over a large enough finite field Fq , in which intermediate network nodes linearly combine their incoming information symbols over Fq, that delivers the information from the sources simultaneously to each receiver at a rate equal to h. An equivalent expression of the main theorem: The source Si transmits symbol σi , which is an element of some finite field Fq. Since each node can linearly combine its inputs, each network link carries a linear combination of its father node inputs.
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