In recent years, network coding ,  has been considered as an auspicious information network paradigm for augmenting the throughput of multiple unicast networks . 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 , 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  and a numerous other papers, for instances , , , , the wave paradigm of information flow was explained and can be illustrated by the following exempli gratia which known as the butterfly network problem.
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 channels, which are represented by the corresponding directed arrows, the throughput is restricted by the premise that it only allows only one message can be dispatched at a time.
Conventionally, information flow in networks are falls within one of two following paradigms: the package paradigm and the liquid paradigm. In the package paradigm, which is almost universally dominated in nowadays information networks (e.g. wireless networks, web-base communication, communication within processors or adopted in processors and external devices communications) [riis], information are considered as blocks which cannot be copied or divided, and the task is to deliver these information packages to a certain destination.
In the liquid paradigm, on the other hand, messages are treated as liquid which means that they can be divided and delivered through different routes before reaching the terminals. One application of this approach is vectors routing . The advantage of this approach is that it allows spreading out and distributing messages over different routes of the network.
For both paradigms, since it has no purpose on delivering data through l1 and l2, all the messages x, y then must be transmitted using the public channel...