# Distributed Satellite Relay Cooperative Communication with Optimized Signal Space Dimension
## 2 Materials and Methods
### Cooperative Transmission of Spatial Information Network  
#### 2.2.1 Universal Multicast Transmission Strategy Based on Ring Planning  
The existing research results basically focus only on a specific model. However, in reality there are more hybrid transmission models. Due to the change of channel transmission model, the signal receiving model of the whole system changes. Optimization criteria are also different. In practical application scenarios, the channel quality is different due to the different geographical location and path loss of different nodes. In order to maximize the use of channel capacity, communication nodes adaptively choose different code rates and modulation methods according to the channel quality. Thus, an asymmetric transmission channel is formed. Therefore, aiming at the complex communication scenario of asymmetric/symmetric coexistence, a generalized method to systematically solve the heterogeneous information transmission under the condition of multicast is proposed. Message transmission is expressed by adjacency of nodes in a weighted directed graph. Each directed edge represents a message flow. The degrees of freedom of the message are represented by the weights of the edges in the directed graph, and the directed edge is omitted when the weight is 0. Take a 4-node multicast channel as an example. Assuming that the number of relay antennas \(N\) is 6, DoF is a tuple \(\mathbf{d}=(3,0,0,1,2,1,1,1,0,1,0,0)\). The information interaction among nodes is shown in Figure 2. If the information exchange occurs at nodes 1 to 2, 2 to 3, and 3 to 1, it is said to constitute a closed loop \(1\to 2\to 3\to 1\). For \(K\)-node multicast relay channels, there may be a closed loop with length of 2 (\(1\to 2\to 1\)) to a closed loop with length of \(K\) (\(1\to 2\to\cdots\to K\to 1\)). As an example, it can be seen in Figure 2a, there is a ring with a length of 2 (\(1\to 2\to 1\)) and (\(2\to 3\to 2\)). There is a ring of length 3 (\(1\to 2\to 3\to 1\)) and (\(1\to 2\to 4\to 1\)). One-way transmission: When the one-way strategy is adopted, the relay uses decoding and forwarding to realize the one-way data communication of the point-to-point channel. The transmitted signal occupies a one-dimensional resource of the relay signal space. If this transmission method is used to realize the message transmission with tuple \(\mathbf{d}\) in DoF, \(d_{\sum}=d_{12}+d_{13}+d_{14}+d_{21}+d_{23}+d_{24}+d_{31}+d_{32}+d_{34}+d_{41 }+d_{42}+d_{43}\leq N\) must be satisfied. However, since \(d_{\sum}=10>N\), the condition \(\mathbf{d}\) cannot be obtained for one-way transmission at this time. Two-way transmission: There is a problem with using a bidirectional strategy to solve 2-ring. The one-dimensional signals \(W^{[1,2]}\) and \(W^{[2,1]}\) interacted by nodes 1 and 2 are aligned to the same signal space of the relay with the signal space alignment (SSA) algorithm. Then, relay can decode to obtain a linear combination of the two signals \(L(W^{[1,2]},W^{[2,1]})\). After the physical layer network coding, this information is sent to nodes 1 and 2 at the same time in the BC phase. Each node uses the message sent by itself as side information to decode the desired signal. After \(1\to 2\to 1\) and \(2\to 3\to 2\) transmissions are implemented using the above method, the remainder of the DOF tuples become \(\mathbf{d}^{\prime}=(d_{12}-d_{21},0,0,0,d_{23}-d_{32},d_{24},d_{31},0,\)\(0,d_{41},0,0)\). If one-way transmission is used to complete the rest of the information interaction, \(d_{12}-d_{21}+d_{23}-d_{32}+d_{24}+d_{31}+d_{41}\) dimensional space is required. Synthesize the above two steps to obtain DOF tuple \(\mathbf{d}\), which needs to meet \(d_{\Sigma}=d_{21}+d_{12}-d_{21}+d_{32}+d_{23}-d_{32}+d_{24}+d_{31}+d_{41}\leq N\). However, due to \(d_{\Sigma}=8>N\), although the use of bidirectional transmission cuts down the two-dimensional signal space requirements, \(\mathbf{d}\) is still not available.