In recent years, network coding [1], [2] has been considered as an auspicious information network paradigm for increasing the throughput of multiple unicast networks [5]. Pioneering research on network coding was undertaken by R. Ahlswede, N. Cai, S.-YR Li and RW Yeung. Their discovery, which was first introduced in [1][2], is considered a crucial advance in modern information theory and the time of its appearance is recognized as the beginning of a new theory , the theory of network coding. In these elegant and succinct papers, within the framework of rigorous mathematics, the glimmer of an optimal network protocol for multi-unicast networks has been introduced in which the key idea is to consider digital information as a wave [riis] . This statement is an advantage in information theory. since it makes a complete distinction between digital information and ordinary products [riis, max Equational logic] which, subsequently, can be used to increase the throughput, scalability and efficiency of information networks. []In [6] and many other papers, e.g. [14], [3], [18], [8], the wave paradigm of information flow has been explained and can be illustrated by the example next, known as the butterfly network problem. Figure 1 Suppose messages x are to be delivered from node i1 to node o1 and message y is to be delivered from node i1 to node o2 in the butterfly network. The messages x and y are considered to be selected characters from a finite set of alphabet named A. The set of alphabetic characters A is assumed to be a finite field, meaning that the result of any finite arithmetic over a finite number d The elements remain in this set. field, for example if x, y  A then x  y  A. For each information channel...... middle of paper ......radigm. In fact, signal superposition (described above) represents an important type of network coding that we will call 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. Some network problems do not have linear solutions, but require the application of nonlinear Boolean functions [18], [8]. Nonlinear network coding has no obvious physical analogue. Rather general network coding represents an information flow paradigm based on a mathematical model where “anything goes”. In this model, there are no a priori restrictions on how information is processed. So in network coding, packets can be copied, opened and mixed. Packet sets can be subjected to very complex nonlinear Boolean transformations.