## 2 Preliminaries and Related Works  
### Network Coding and Its Fundamental Concepts  
Based on Figure 19.3 in [27], we give an enhanced version of this sketch map and many new elements are added. Using the new Figure 1, we will illustrate some important concepts of network coding about NEC which are referred to in this paper. These concepts are necessary for understanding the following Algorithm 1. Some related works are also referred to and a short introduction about them will be given based on Figure 1. Because of space limitation, we will not pursue a strict definition and instead give a descriptive statement. The precise definitions of concepts are illustrated in the referred background papers. Figure 1 is the famous butter-fly picture in the research field about network coding. If the network is error-free, the source node 1 wants to multicast a message vector \(X=\left[X_{1}\ X_{2}\right]^{T}\) to both sinks 6 and 7. Some concepts are list below. Dimension of network coding: It is equal to the max flow min cut of the multicast network. In Figure 1, the dimension denoted as \(w\) which is the size of \(X\), therefore 2. Coding field: The coefficients of network coding are selected from the finite field which is denoted as \(F_{q}\) whose size is \(q\). Variables \(a\), \(b\),..., \(l\), which are coding coefficients, take value from \(F_{q}\). All the messages including \(u\), \(X\), \(Y\), \(Z\), and all the messages in the relay nodes take value in an extension field \(F_{Q}\) with size \(Q\). \(G\) takes value in \(F_{Q}\) while \(T\) and \(T_{Z\to Y}\) take values in \(F_{q}\). Local coding kernel: A feasible linear network coding scheme consists of a scalar \(k_{d,e}\), called the local encoding kernel, for every adjacent \((d,e)\). The \(\left|In(v)\right|\times\left|Out(v)\right|\) matrix \(K_{v}=\left[k_{d,e}\right]_{d\in In(v),e\in Out(v)}\) is called the local encoding kernel at node \(v\). \(K_{s}=\left[ \begin{array}{cc}a&c\\ b&d\end{array} \right]\), \(K_{t}=\left[ \begin{array}{cc}f&e\end{array} \right]\), \(K_{u}=\left[ \begin{array}{cc}g&h\end{array} \right]\), \(K_{u}=\left[ \begin{array}{cc}i\\ j\end{array} \right]\), \(K_{x}=\left[ \begin{array}{cc}k&l\end{array} \right]\) are all local coding kernels. Global coding kernel: A feasible linear network coding scheme also consists of a column vector \(f_{e}\) for every channel \(e\) such that: (1) The vector \(f_{e}\) for imaginary channels \(e\in In(s)\) form the standard basis of the vector space \(F_{q}^{w}\); (2) \(f_{e}=\sum_{d\in In(v)}k_{d,e}f_{d}\) for \(e\in Out(v)\). Global coding kernel and local coding field are two different mathematical descriptions, i.e., two sides of the same coin. They can be deduced from each other. \(X\cdot f_{e}=X\sum\limits_{d\in In(v)}k_{d,e}f_{d}=\sum\limits_{d\in In(v)}k_{ d,e}(X\cdot f_{d})\). ```
Step 1 Set involved parameters in L1 optimization according Equations (2) and (3). Step 2 In real field \(F_{Q}=R\), add \(n-k\) zeros to \(\kappa\) behind of it, to form \(x_{0}\in R^{n\times 1}\). Get \(y\in R^{(m+n)\times 1}\) based on \(y=\left[ \begin{array}{c}A\\ I_{n}\end{array} \right]\times x_{0}\). The coding matrix is as the matrix in Equation (6). Step 3 In a finite field \(F_{q}\), perform encoding procedure of network coding scheme in every relay node. Step 4 Divide the vector \(y\) into two parts: \(y=\left[ \begin{array}{c}y_{A}\\ y_{B}\end{array} \right]\) where \(y_{A}\in R^{(\lambda\cdot C)\times 1}\) and \(y_{B}\in R^{(m+n-\lambda\cdot C)\times 1}\). To adapt to the transmission through networks, reorganize the vector \(y\in R^{(m+n)\times 1}\) to the matrix \(y\prime\in R^{C\times\frac{m+n}{C}}\) where \(y\prime=\left[y\prime_{1},\ldots y\prime_{j},\ldots y\prime_{\frac{m}{C}} \right](j=1,2,\ldots,\frac{m+n}{C})\) and \(y\prime_{j}\in R^{C\times 1}\)\((j=1,2,\ldots,\frac{m+n}{C})\).