## 2 System Model and Channel Description  
### System Model  
In the SGIN framework presented in Figure 1, NTN elements (satellites and UAVs) are deployed to provide seamless coverage extension for vessels operating in offshore and deep-sea zones where terrestrial BSs are unavailable. We find that the TWRN, configured with one relay node and two source nodes, constitutes the minimal functional unit for enabling multi-hop transmissions. Throughout this manuscript, we take into account a maritime TWRN scenario adopting BMST-coded PNC scheme, where two separated nodes enable long-distance data transmission with the aid of a relay node. PNC takes advantage of the superposition property of signals and converts the collision information caused by simultaneous transmission of two source nodes into network-coded information [27]. Compared to traditional orthogonal transmission, this method achieves information exchange using only half the time resources, thereby doubling the network throughput [28]. We assume both the absence of direct connectivity between nodes and the perfect knowledge of channel state information (CSI). Specifically, the packet switching process comprises two distinct phases: the uplink access (UA) phase and the downlink broadcast (DB) phase. During the UA phase, the signal processing procedure of the proposed system is depicted in Figure 2. At source node \(N_{i}\), the original binary sequence \(\mathbf{u}_{i}=\{u_{i0},u_{i1},\ldots,u_{i2K-1}\}\) is encoded by a linear encoder, yielding a coded sequence \(\mathbf{c}_{i}=\{c_{i0},c_{i1},\ldots,c_{i2N-1}\}\), where \(i\in\{1,2\}\). We assume the coded sequence \(\mathbf{c}_{i}\) is then mapped to a symbol sequence \(\mathbf{x}_{i}=\{x_{i,0},x_{i,1},\ldots,x_{i,N-1}\}\) through the quadrature phase shift keying (QPSK) mapping, in which \(x_{i,r}\in\mathcal{S}=\frac{1}{\sqrt{2}}\{1+j,-1+j,1-j,-1-j\},j=\sqrt{-1}\) and \(0\leq r\leq N-1\). Notably, QPSK is adopted here as a representative modulation scheme, without loss of generality, as the proposed framework supports higher-order modulation formats. Assuming that the modulated symbols from both source nodes arrive at relay node \(N_{R}\) simultaneously. Thus, for node \(N_{R}\), the \(r\)-th received symbol \(y_{r}\in\mathbf{y}=\{y_{0},y_{1},\ldots,y_{N-1}\}\) can be calculated by  
\[y_{r}=h_{1,r}x_{1,r}+h_{2,r}x_{2,r}+w_{r}, \tag{1}\]  
where \(h_{1,r}\) and \(h_{2,r}\) represent the channel gains from the nodes \(N_{1}\) and \(N_{2}\) to the relay node \(N_{R}\), respectively. \(w_{r}\) denotes a complex-valued white Gaussian noise. Upon receiving the symbol vector \(\mathbf{y}\), PNC mapper performs maximum-likelihood (ML) detection and exchanges soft information with two decoders iteratively. After iterative decoding, the estimate version \((\widehat{\mathbf{n}_{1}},\widehat{\mathbf{n}_{2}})\) of the original binary sequence pair \((\mathbf{u}_{1},\mathbf{u}_{2})\) is obtained. The relay node then performs computing (network coding) operation, which can be viewed as  
Figure 1: This is a space-ground integrated network framework. Figure 2: This is the signal processing procedure for UA phase of channel-coded PNC. a linear mapping from \(\mathbb{F}_{2}^{2}\) to \(\mathbb{F}_{2}\)[29]. Finally, the estimated network-coded sequence \(\widehat{\mathbf{u}_{R}}\) can be expressed as  
\[\widehat{\mathbf{u}_{R}}=\widehat{\mathbf{u}_{1}}\oplus\widehat{\mathbf{u}_{2}}. \tag{2}\]  
Under ideal conditions, \(\widehat{\mathbf{u}_{R}}\) is equivalent to \(\mathbf{u}_{R}=\mathbf{u}_{1}\oplus\mathbf{u}_{2}\). During the DB phase, the relay node \(N_{R}\) re-encodes the obtained sequence \(\mathbf{u}_{R}\) and broadcasts it to both source nodes. Knowing its own information \(\mathbf{u}_{1}\), the source node \(N_{1}\) can extract the binary sequence \(\mathbf{u}_{2}\) transmitted by node \(N_{2}\) from the network-coded sequence \(\mathbf{u}_{R}\). This process can be described as an XOR operation, i.e., \(\mathbf{u}_{1}\oplus\mathbf{u}_{R}=\mathbf{u}_{2}\).