Network Coding

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In recent years, network coding [1], [2] has been considered as an auspicious information network paradigm for augmenting the throughput of multiple unicast networks [5]. The pioneering researches of network coding were undertaken by R. Ahlswede, N. Cai, S.-Y.R. Li and R.W. Yeung. Their discovery, which was first introduced in [1][2], are considered to be the crucial breakthrough in modern information theory and the time of its appearance, is recognized as the beginning of a new theory-Network Coding theory. In these elegant, succinct articles, within the purview of rigorous mathematics, the glimmering of an optimal network protocol for multiple unicast network was introduced in which the key idea is considering digital information as wave [riis].

This contention is an advantage in information theory since it fully distinguishes between digital information and ordinary commodities [riis, max Equational logic] which, in follows, can be used for boosting throughput, scalability and efficiency of information networks.[]

In [6] and a numerous other papers, for instances [14], [3], [18], [8], the wave paradigm of information flow was explained and can be illustrated by the following exempli gratia which known as the butterfly network problem.

Figure 1

Suppose that the messages x has to be transmitted from node i1 to node o1 and message y must be delivered from node i1 to node o2 in the butterfly network. The messages x and y are considered as characters selected from a finite set of alphabet named A. The alphabet characters set A are assumed to be a finite field, which means that the outcome of any finite arithmetic on a finite numbers of elements remains in that field, e.g. if x, y  A then x  y  A.

For each information chann...

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...radigm. In fact, super-positioning of signals (described above) represents an important type of Network Coding we will refer to as Linear Network Coding (see also [15]). Although Linear Network Coding represents a very important subclass of Network Coding, in general Network Coding involves methods that go beyond linear Network Coding. Certain network problems have no linear solutions, but require the application of non-linear boolean functions [18], [8]. Non-Linear Network Coding has no obvious physical analogue. Rather general Network Coding represents a paradigm of information flow based on a mathematical model where 'every thing goes'. In this model there are no apriory restrictions on how information is treated. Thus in Network Coding, packets might be copied, opened and mixed. Sets of packets might be subject to highly complex non-linear boolean transformations.

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