## 3 Improve L1 Optimization to Correct 100% of Corrupted Propagated Errors in Network Coding
### An Example about Algorithm 1  
To aid easy understanding of Algorithm 1, we give a specific example. The most important aspect is that some parameters are set far smaller than its own real value in Algorithm 1 for the limited space. However, as an example, the essence of it is the same with Algorithm 1 though some parameters are smaller than the real value. Step 1 Set involved parameters, among them, \(m=16\), \(n=4\), \(C=4\). Step 2\(\kappa\ =\ \left[ \begin{array}{c}1\\ 1\end{array} \right]\), \(x_{0}\ =\ \left[ \begin{array}{c}1\\ 1\\ 0\end{array} \right]\), \(y\ =\ \left[ \begin{array}{c}A\in R^{16\times 4}\\ I_{4}\end{array} \right]\times x_{0}\ =\ \left[ \begin{array}{c}A\times x_{0}\\ I_{4}\times x_{0}\end{array} \right]\ =\ \left[ \begin{array}{c}A\times x_{0}\\ x_{0}\end{array} \right]\). Here,  
\(\left[ \begin{array}{c}A\in R^{16\times 4}\\ I_{4}\end{array} \right]\) is the trap matrix. \(y\ =\ \left[1.012,\,0.986,\,1.101,\,1.014,\,1.052,\,1.022,\,1.024,0.991,\,0.957,\, 0.994,\,1.003,\,0.994,\,1.017,\,0.990,\right.\)  
Step 3 In a finite field \(F_{q}\), perform encoding procedure of network coding scheme in every relay node. The corresponding transfer matrixes are: the message transfer matrix \(\hat{T}\), the error message transfer matrix \(T_{Z\to Y}\), and the error head vector transfer matrix \(L\). Step 4 Reorganize the vector \(y\ \in\ R^{(m+n)\times 1}\) to the matrix \(y^{\prime}\ \in\ R^{C\times\frac{n+n}{C}}\), \(y^{\prime}\ =\ \left[ \begin{array}{cccc}1.012&1.052&0.957&1.017&1.007\\ 0.986&1.022&0.994&0.990&0.997\\ 1.101&1.024&1.003&1.006&1.016\\ 1.014&0.991&0.994&1.007&1.023\\ \end{array} \right]\). Divide the matrix \(y^{\prime}\) into two parts: \(y^{\prime}\ =\ \left[y^{\prime}{}_{A}\ ;\ y^{\prime}{}_{B}\right]\). \(y^{\prime}{}_{A}\ =\ \left[ \begin{array}{cccc}1.052&0.957&1.017&1.007\\ 1.022&0.994&0.990&0.997\\ 1.024&1.003&1.006&1.016\\ 0.991&0.994&1.007&1.023\\ \end{array} \right]\). Set \(\lambda\ =\ 1\). Send the matrix \(y^{\prime}{}_{A}\ =\ \left[ \begin{array}{c}1.012\\ 0.986\\ 1.101\\ 1.014\\ \end{array} \right]\) through a secret channel. Send the matrix \(y\prime_{B}=\left[ \begin{array}{cccc}1.052&0.957&1.017&1.007\\ 1.022&0.994&0.990&0.997\\ 1.024&1.003&1.006&1.016\\ 0.991&0.994&1.007&1.023\\ \end{array} \right]\) through the networks where the networks coding is performed in the relay nodes. * \(y\prime_{B}=\left[ \begin{array}{cccc}1.052&0.957&1.017&1.007\\ 1.022&0.994&0.990&0.997\\ 1.024&1.003&1.006&1.016\\ 0.991&0.994&1.007&1.023\\ \end{array} \right]\), the result of network coding in relay nodes is expressed by the equation \(Y_{j}=\hat{T}\cdot y\prime_{j}+T_{Z\to Y}\cdot\left(Z-L\cdot y\prime_{j} \right)\)\((j=2,3,4,5)\) where \(\hat{T}\) is known by the coding vector in the head of packets, but \(T_{Z\to Y}\) and \(L\) are all unknown. \(Y_{B}=\left[ \begin{array}{cccc}Y_{2},\ Y_{3},\ Y_{4},\ Y_{5}\end{array}  \right]=\left[ \begin{array}{cccc}1.123&0.976&0.817&1.013\\ 1.102&0.983&0.978&0.998\\ 1.012&1.012&1.012&1.023\\ 0.979&0.972&1.023&1.011\\ \end{array} \right]\) are received messages in the sink. * Perform network coding decoding algorithm to \(Y_{B}\). The result of network coding decoding is expressed by the equation \(\left(y\prime_{j}\right)^{d}=\hat{T}^{-1}\cdot Y_{j}\)\((j=2,3,4,5)\) where \(\hat{T}^{-1}\) is known. Because \(T_{Z\to Y}\cdot\left(Z-L\cdot y\prime_{j}\right)\) is unknown, we cannot get \(y\prime_{j}=\left(Y_{j}-T_{Z\to Y}\cdot Z\right)/\left(\hat{T}-T_{Z\to Y} \cdot L\right)\)\((j=2,3,4,5)\). Thus, we let \(\left(y\prime_{j}\right)^{d}=\hat{T}^{-1}\cdot Y_{j}\)\((j=2,3,4,5)\) as the estimate of \(y\prime_{j}=\left(Y_{j}-T_{Z\to Y}\cdot Z\right)/(\hat{T}-T_{Z\to Y} \cdot L)\)\((j=2,3,4,5)\). Then we will perform decoding with L1 optimization to get \(y\prime_{j}\) based on \(\left(y\prime_{j}\right)^{d}\). \(\left(y\prime_{B}\right)^{d}=\left[\left(y\prime_{2}\right)^{d},\ \left(y\prime_{3}\right)^{d},\ \left(y\prime_{4}\right)^{d},\ \left(y\prime_{5}\right)^{d}\right]= \left[