## 3 Physical-Layer Network Coding Based on Block Markov Superposition Transmission
### Iterative Decoding and Computing Algorithm
This decoding algorithm operates on a subgraph comprising \(d+1\) consecutive layers, as described in Algorithm 3. For an RSPC with a code length of \(n(n-2)\) and a code rate \(k/n\), each decoding layer in the normal graph has \(n(n-2)\) parallel nodes \(\boxed{\pi}\) of degree \(m+2\), \(n(n-2)\) parallel nodes \(\boxed{+}\) of degree \(m+2\), and a node of type \(\boxed{\pi}\). Since both the node \(\boxed{+}\) and the node \(\boxed{=}\) have computational complexity \(O(m+2)\), the decoding complexity varies depending on the baisc code. As mentioned in [33], the computational complexity is linear with the code length. For the decoding algorithm, the actual computational loads depend on both the decoding delay and the number of iterations. Further technical details and theoretical analysis can be found in [36]. To mitigate the loss of useful information during the decoding process, we strategically implement network coding operations after each iterative decoding stage. In the proposed BMST-RSPC-PNC system employing dual decoders, the iterative decoding process forms a closed-loop architecture. This architecture can be formally characterized by the following cyclic sequence: PNC mapper\(\rightarrow\)Decoder 1\(\rightarrow\)PNC mapper\(\rightarrow\)Decoder 2\(\rightarrow\)PNC mapper\(\rightarrow\cdots\rightarrow\)PNC mapper. The detailed implementation of the corresponding DC algorithm is systematically presented in Algorithm 4. While inter-decoder iteration introduces additional computational complexity, this mechanism effectively preserves critical information integrity at relay node, leading to measurable performance enhancements. cation over a channel. [PERSON] has demonstrated that the error probability can be made arbitrarily small for all communication rates below the channel capacity. To achieve reliable communication, it is essential to understand the capacity of both links during the UA phase. According to the chain rule, the joint mutual information between the transmitted symbol pair \((\mathbf{x}_{1},\mathbf{x}_{2})\) and the received superposition signal \(\mathbf{y}\), conditioned on the channel gains \((\mathbf{h}_{1},\mathbf{h}_{2})\), can be expressed as  
\[ \begin{split} I(\mathbf{x}_{1},\mathbf{x}_{2};\mathbf{y}|\mathbf{h}_{1},\mathbf{h}_{2} )&=I(\mathbf{x}_{1};\mathbf{y}|\mathbf{h}_{1},\mathbf{h}_{2})+I(\mathbf{x}_{2};\mathbf{y}| \mathbf{x}_{1},\mathbf{h}_{1},\mathbf{h}_{2})\\ &=I(\mathbf{x}_{2};\mathbf{y}|\mathbf{h}_{1},\mathbf{h}_{2})+I(\mathbf{x}_{1};\mathbf{y}| \mathbf{x}_{2},\mathbf{h}_{1},\mathbf{h}_{2}),\end{split}  \tag{9}\]  
which, denoted as \(C\), is the theoretical upper bound on the total spectral efficiency during the UA phase. Specifically, the mutual information between the transmitted symbol variable \(\mathbf{x}_{1}\) and the received superposition variable \(\mathbf{y}\) can be decomposed as  
\[ \begin{split} I(\mathbf{x}_{1};\mathbf{y}|\mathbf{h}_{1},\mathbf{h}_{2})& =H(\mathbf{x}_{1})-H(\mathbf{x}_{1}|\mathbf{y},\mathbf{h}_{1},\mathbf{h}_{2})\\ &=\log_{2}(|\mathcal{S}|)+\mathbb{E}[\log_{2}\Pr\{\mathbf{x}_{1}|\mathbf{ y},\mathbf{h}_{1},\mathbf{h}_{2}\}],\end{split}  \tag{10}\]  
and  
\[ \begin{split} I(\mathbf{x}_{1};\mathbf{y}|\mathbf{x}_{2},\mathbf{h}_{1},\mathbf{h}_{2 })&=H(\mathbf{x}_{1})-H(\mathbf{x}_{1}|\mathbf{y},\mathbf{x}_{2},\mathbf{h}_{1},\mathbf{h }_{2})\\ &=\log_{2}(|\mathcal{S}|)+\mathbb{E}[\log_{2}\Pr\{\mathbf{x}_{1}|\bm {y},\mathbf{x}_{2},\mathbf{h}_{1},\mathbf{h}_{2}\}].\end{split}  \tag{11}\]  
Here, \(H(\cdot)\) denotes the entropy, and \(|\mathcal{S}|\) represents the number of constellation points in \(\mathcal{S}\), respectively. The \(\mathbb{E}[\cdot]\) can be approximated via Monte Carlo simulation by averaging over channel realizations and noise samples. In the marine TWRN system, we assume equal transmission power for both senders.