## 3 Improve L1 Optimization to Correct 100% of Corrupted Propagated Errors in Network Coding
### The Transfer Model in Non-Coherent Network  
In the introduction we described the transfer model in the coherent network. The transfer model in the non-coherent network is different from the transfer model in the coherent network. We should clarify this model in detail because it is important for the network coding decoding. A classical random network code indicates that \(y\) includes the identity matrix as a part of each batch. The identity matrix sent by source experiences the same transform matrix \(T\) with the raw data of the batch. Thus,  
\[\hat{T}=\ T\cdot I+T_{Z\to Y}\cdot L \tag{9}\]  
where \(\hat{T}\) and \(L\) are the columns corresponding to \(I\)'s location in \(Y\) and \(Z\) respectively. \(\hat{T}\) is \(C\times C\), and \(L\) is \(t\times C\). By substituting \(T\) into Equation (9), Equation (8) can be simplified as:  
\[Y=\hat{T}\cdot y+T_{Z\to Y}\cdot(Z-Ly) \tag{10}\]  
Note that the matrix \(\hat{T}\) acts as a proxy transfer matrix for \(T\), which the sink does not know. Note that the above is mainly in reference to [25]. Equation (10) is slightly different from \(Y=T\cdot y+T_{Z\to Y}\cdot Z\) which is for the coherent network. Equation (10) is for random network coding. In random network, \(T\) is unknown and it is replaced by \(\hat{T}\). \(Y=T\cdot y+T_{Z\to Y}\cdot Z\) is degraded from Equation (10) for random networks. In the coherent networks, there is no error in header because there are no coding vectors in the head of the packets. Therefore, \(L\) is 0 matrix.  
In the sink, packets are collected until the proxy transfer matrix (\(\hat{T}\)) is invertible. Matrix \(\hat{T}^{-1}\) is left multiplied in Equation (10), we get  
\[y=\hat{T}^{-1}\cdot Y-\hat{T}^{-1}\cdot T_{Z\to Y}\cdot(Z-Ly) \tag{11}\]  
where \(\hat{T}^{-1}\cdot Y\) can be got, and \(\hat{T}^{-1}\cdot T_{Z\to Y}\cdot(Z-Ly)\) is unknown. Let \(y^{d}=\hat{T}^{-1}\cdot Y\). \(y^{d}\) is the result of network coding decoding in the sink. \(y^{d}\) can be regarded as a deviation value of \(y\). In principle, \(\hat{T}\) can be seen as a proxy transfer matrix of the true transfer matrix \(T\) to perform decoding.  
However, there is a difference of \(\hat{T}^{-1}\cdot T_{Z\to Y}\cdot(Z-Ly)\) between \(y^{d}\) and \(y\). The difference need to be corrected through L1 optimization in [21], rather than the decoding algorithm of the traditional code. Above all, the number of \"original error\" is \(Z\), and the number of \"propagated error\" is \(\hat{T}^{-1}\cdot T_{Z\to Y}\cdot(Z-Ly)\). The Errors in header mentioned above is expressed by \(L\). \(\hat{T}^{-1}\cdot T_{Z\to Y}\cdot(Z-Ly)\), which is \(C\times 1\), represents the spread result of \(Z\). In random network coding, we just know \(\hat{T}^{-1}\). However, \(T_{Z\to Y}\), \(Z\), \(L\) and \(y\) are all unknown. Theoretically, even if \(t\), the number of original corrupted packets, is very small, \(\hat{T}^{-1}\cdot T_{Z\to Y}\cdot(Z-Ly)\) also has potential to have \(C\) nonzero components. That is to say, \(\hat{T}^{-1}\cdot T_{Z\to Y}\cdot(Z-Ly)\) pollutes every symbol of the messages \(y^{d}\).  
With \(wr(\beta)\) denote the number of nonzero components (or symbols) in an arbitrary vector or matrix \(\beta\). With \(wr_{norm}(\beta)\in[0,1]\) denote the normalized \(wr(\beta)\). If \(wr_{norm}\big{(}\hat{T}^{-1}\cdot T_{Z\to Y}\cdot(Z-Ly)\big{)}\) is 1, where the percentage of propagated errors is 100%, we cannot decode successfully with [21]. If \(wr_{norm}\big{(}\hat{T}^{-1}\cdot T_{Z\to Y}\cdot(Z-Ly)\big{)}\) is high, for example, 0.99999, [PERSON]'s Model can decode successfully with a large \(m\)[21]. However, the information rate is very low. The above statement accords with the truth: the more errors there are, the lower the information rate is. Any method cannot contradict this basic truth. Therefore, in random networks, we can just control the sparseness of \(\hat{T}^{-1}\) partly.