## 1 Introduction
If network coding theory is to be applied into practice from the theory in laboratory, it is critical to find a new error correction mechanism to cope with the propagated error in network coding. It is time to think outside the constraints of the current error correction mechanism to find brand new thinking from other research fields. Recently, researchers have made great progress in the field of social networks and L1 optimization and sparse learning [19; 20]. Based on L1 optimization, [PERSON] and [PERSON] propose a dense error correction method which can correct nearly 100% of the corrupted observations which may seem surprising and unbelievable at first observation [21]. The powerful error correcting ability of [21] naturally reminds us to introduce it to correct the dense propagated error in network coding. There are also some works applying the theory of social network to improve the performance of WSN [22; 23; 24]. With the two emerging techniques, hopefully, we will arrive at a solution to the thorny problem of error propagation in network coding. Although [PERSON] and [PERSON]'s scheme looks as though it could solve the error propagation problem, it fails in the last mile. In addition, although [PERSON] and [PERSON]'s L1 optimization model can correct dense error where nearly (not exactly) 100% of the corrupted observations can be corrected, it cannot correct the propagated errors in network coding which usually pollute exactly 100% of the received messages. If we can bring down the fraction of propagated error to a little below 100%, this problem goes away. However, it is not a simple matter of reducing the fraction of error propagation in the network. The works as [11; 12; 13; 14] are the most important works about NEC, but none of them can reduce the fraction of propagated errors. By introducing a secret channel and a matrix which can trap some errors, we successfully decrease the fraction of propagated error to a little less than 100%. The first method is using a secret channel to transmit a small part of messages in advance which will indirectly bring down the fraction of propagated error to a little less than 100%. The secret channel method requires simple and straightforward thinking, but it is very effective. The secret channel will not cost a lot of resources [25] because the percentage of messages which need to be sent can be very low, for example, 1%. As long as the faction of ultimate errors are not equal to 100%, [PERSON] and [PERSON]'s scheme can work. The second method is to set the bottom of the coding matrix in [PERSON] and [PERSON]'s scheme as an identity matrix. The original uncompressible message is added with many zeroes in the bottom part to form a sparse message vector. The sparse message is then coded with a coding matrix which is constructed based on [PERSON] and [PERSON]'s scheme. Because we know a priori that the bottom part of the sparse message vector is all zeroes, we can trap a part of errors from the received messages. This will indirectly decrease the rate of the corrupted messages. With the two novel methods, [PERSON] and [PERSON]'s L1 optimization model successfully solves the error propagation problem in network coding. Although [PERSON] and [PERSON]'s scheme can ultimately solve the error propagation problem after our two improvements, its fraction of successful decoding is too low while the fraction of errors is in the range of between 0.85 to 1. If the received messages are not \"informative\" enough, the severe trend could worsen. In the context of network coding whose decoding algorithm is solving equations other than L1 optimization, the statement that \"the received messages are informative\" refers to the famous \"all-or-nothing\" problem [26]. In a sink, if the \(C\) received messages are full rank, we can decode successfully based on the method of solving equations, and we call the received messages \"informative\" or \"innovative\".