## 1 Introduction
The sink cannot recover any information from received data, unless it receives at least the same number of innovative packets as were originally combined together. However, in the context of network coding whose decoding algorithm is L1 optimization other than solving equations, the \"all-or-nothing\" problem will be partially alleviated. Even if the received packets are not full rank, there is also a potential to decode the original error with L1 optimization rather than solving the equation. The property of alleviating the \"all-or-nothing\" problem in our scheme is similar to that of [26], referring to the compress sensing. However, the more the rank of received packets is, the greater the opportunity for the L1 optimization to decode successfully. The objective is to have relay nodes receive more \"informative\" or \"innovative\" packets from upstream nodes. However, it is not an easy task in the random environment to receive more \"informative\" packets. For a relay node, the definition of \"informative\" in the context of L1 optimization does not merely refer to having the full rank as far as possible. Even if all the relay nodes will transmit packets with full rank to the downstream nodes, the received packets are not exactly full rank. Providing the full rank of packets from upstream nodes is a local optimum solution other than the global optimum solution. If some upstream nodes whose packets not only are full rank for themselves but also make multi-hops away downstream nodes have full rank of packets, we say such upstream nodes are more \"informative\". We can use one acknowledgement message from multi-hops away downstream nodes to identify which upstream nodes are more \"informative\". The point is that we cannot use acknowledgement messages all the time due to their high resource consuming nature. The reputation-based trust model of the social network research field will help us to find the upstream nodes that have more opportunities to be informative for any time other than one time. If a relay node in network coding, identifies which upstream nodes that can bring more \"informative\" packets in advance, the received packets by the sink will have more opportunities to be full rank. The more acknowledgement characters (ACKs) received by a relay node, the more the trust value is. After the trust computation stage, we can select the optimal relay nodes to perform network coding, and other relay nodes will not perform network coding to save energy. Selecting relay nodes with high trust value will help the received packets of the sink node to be full rank, therefore L1 optimization will have more opportunities to decode successfully. The main contribution of this paper is:* Many 0 values are added to the original information to make the signal sparse; re-organize the transmitted signal from vector to matrix. The two methods allow L1 optimization method to be applied in network coding; therefore, L1 optimization technique can be introduced from the research field of image recognition to the research field of communication. * The coding matrix in the original model by [PERSON] and [PERSON] is replaced by a specially designed matrix in which the bottom is identity matrix. The specially designed matrix can trap some errors and make these trapped errors known by the sink. The method will indirectly put down the fraction of propagated errors a little because we can know some errors a prior through the trapped errors. * We use a secret channel to transmit a small amount messages in advance which will indirectly bring down the fraction of propagated errors slightly below 100%. Based on this method, [PERSON] and [PERSON]'s model can correct propagate errors in network coding. * We propose a new distributed approach that establishes reputation-based trust among sensor nodes in order to identify the informative upstream sensor nodes. This will help L1 optimization have more opportunities to decode successfully and it will result in short delays and high throughputs. The remainder of this paper is organized as follows.