Application of Groebner Bases to Study a Communication System

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.


Introduction:
Communication is the exchange of information between individuals by different means of transmission. The simplest communication system can consist of an information source, and a receiver and the link between them is called a communication channel, which can be a wire or wireless or the air range in which the electromagnetic waves propagate between the source and the receiver.
A communications network consists of a set of source nodes. Each node generates a symbol or set of symbols taken from a finite field, as well as a set of downstream nodes, in addition to a set of internal nodes. These nodes are linked to each other through a set of channels so that each channel transmits a specific amount of data called the channel capacity. 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.
Robertet al.found that it is possible to find a solution to the network coding problem using linear coding, but 3 R. Dougherty, C. Freiling, and K. Zeger proved that the linear coding was not sufficient to solve the problem, so nonlinear coding was used, and several studies have shown the importance of this type of coding in information theory.
Network coding is an area of research created in papers from the late 1990s to early the second millennium 4 5 . However, the concept of network cipher, and especially linear network encryption, appeared much earlier in 1978 6 The communication network contains a set of source nodes so that each node generates a symbol from its finite field. These symbols are called data units, and it also includes a set of downstream nodes, a set of internal nodes. These nodes are linked with each other through a group of channels so that each channel transmits a specific amount of data called channel capacity.
If the communications network contains a single source node and a set of downstream nodes that are asking for data generated in this source node, then the connection problem over this network is called a multicast transmission 7 while, it is called Intersession network coding 8 if it contains two source nodes and two downstream nodes such that each downstream node requests the symbols generated in one of the source nodes.
The coding problem is solvable if and only if all the target nodes can get the message using only the information they received; otherwise, it is not solvable. ( 1 , … , ) = 1 1 + ⋯ + ∶ ∈ . Definition 2: 9 The Jacobian polynomial of ( , ), ( , ) with coefficients from a field is the determinant of the form: Definition 3: 9 The ideal of a ring ℛ is defined as a non-empty subset that achieves: 1-is an additive subgroup of ℛ with (+).
2. Whatever ∈ and ∈ ℛ then . ∈ . Definition 4: 9 For ⊆ ℛ a non-empty subset, the ideal generated by a set has the form: = {∑ 1 ∶ ∈ ℛ, ∈ }. If = { 1 , … , } is a finite set, then the ideal is finitely generated and write = 〈 1 , … , 〉. Also, the ideal is generated by the set ⊆ ℛ, which can be expressed as the intersection of all ideals in ℛ. Each of them contains the ideal .

Definition 6: 9
Let be a field, and let , ℎ, 1 , … , ∈ [ 1 , … , ] be polynomials where ≠ 0, (1 ≤ ≤ ) and = { 1 , … }, then is reduced to ℎ (via ), and denoted by → + ℎ if and only if there is a sequence of indexes 1 , … , ∈ {1, … , } and a sequence of polynomials ℎ 1 , … , ℎ −1 ∈ [ 1 , … , ] such that: ii. The message is generated in the source and must be transferred to all target nodes in . Also an encoding function related to link = ( , ) is defined as follows: Where ( ) is the set of links entering the node . Put ( , ) ;1 ≤ ≤ ℎ, the ℎ edge-disjoint paths from the sources to receiver , 1 ≤ ≤ . Links will be carrying linear combinations of their father node inputs, and the set { } denotes the coefficients used in these linear combinations. Put to refer to the symbol on the last link of the path ( , ). Therefore, receiver has to solve the following system of equations: Where are ℎ × ℎ matrices which are the receiver transfer matrices 12 .
Note that the elements of are polynomials in { }.

Example 2: 12
Consider a network with two sources and three receivers, as in (Fig. 1). Note that there is two edge disjoint paths from the sources to each receiver (Fig. 1a). Therefore, each receiver can receive the information from both sources when using the network alone. However, when all three receivers use the network at the same time, then the intersections between paths at BD and GH have to be resolved. 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 . 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. Then there exists a multicast transmission scheme over a large enough finite field , in which intermediate network nodes linearly combine their incoming information symbols over , 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 transmits symbol , which is an element of some finite field . Since each node can linearly combine its inputs, each network link carries a linear combination of its father node inputs. So, links carry linear combinations of source symbols , and a receiver can recapture the source information if the ℎ links it observes carry independent linear combinations of the .

Theorems
This paragraph presents our contributions to find the necessary and sufficient conditions under which the nodes can combine their inputs which guarantee the ability to solve the problem of multicast transmission.   for 1 ≤ ≤ : ( ) = . For every 1 ≤ ≤ , which means that is the opposite of (not necessarily polynomial in the general case). To prove the independence, suppose as in (theorem 4 ) the existence of ℎ ∈ [ ] such that ℎ( 1 , … , ) = 0. Then 0 = ℎ( 1 ( ), … , ( )) = ℎ( 1 , … , ) = ℎ, which means that the polynomials 1 , . . , are algebraically independent of .

Simplify communications network
The source sends a copy of the data. It generates to each of the downstream nodes. (Fig. 2) shows the transmission of the symbols 1 , 2 from the source nodes to the target node 1 , 2

Figure 2
Construction algorithm 1-Choose the channels so that the same amount of data will flow through them 2-Form partial networks so that the flow through the channels of each sub network is the same amount of data 3-Represent every partial network by two nodes, and a channel whose capacity is the available capacity of the channels of the considered subnetwork 4-Call the node at the beginning of the channel, a distributor (HOP). Note: assume that every HOP distributor receives messages from the source node and can process and forward the messages. 5-Represent the set of channels in the original network that connects a node from one subnet to a node from another subnet, with a channel from the node in the first subnet to the distributor in the second subnet (and that is in the new network).

Application 1:
Applying the algorithm to the previous network in (Fig. 2), gives the network that is shown in (Fig. 3). Consider the network as in (Fig. 4)

Figure 4
The set of polynomials that express the distributor node are as follows = { 1 − 3 , 2 − 4 , 3  (Theorem 6 proves that this basis always has this shape). The new network corresponding to the Groebner basis will be as in (Fig. 5). Where 1 . 1 = 2 . 2 then ( 1 , 2 ) = − 1 . 1 + 2 . 2 . Which is either a zero polynomial or from the desired form, also the reduction maintains this form, this is a direct consequence because reduction through polynomials of the desired form is similar