Dual-Domain Superposition for Maritime Relay Communications: A Flexible-Coded Transmission Design Towards Spectrum–Reliability Synergy

Abstract

Maritime relay communication has emerged as a critical application scenario for non-terrestrial networks (NTNs), providing beyond-line-of-sight (BLOS) connectivity for offshore terminals. Unlike terrestrial environments, the complex marine propagation conditions lead to signal instability. To enhance the robustness of maritime two-way relay networks (TWRNs), we propose a novel physical-layer network coding (PNC) scheme based on block Markov superposition transmission (BMST). The proposed scheme introduces a novel co-design framework that achieves dual breakthroughs: (1) robust error correction via BMST’s spatially coupled coding architecture and (2) spectral efficiency maximization through PNC’s spatial-domain signal superposition. Moreover, we develop a decoding–computing (DC) algorithm that sequentially performs iterative decoding followed by computing. Compared to the computing–decoding (CD) algorithm, the proposed DC algorithm mitigates useful information loss at relay nodes, achieving a 2.9 dB coding gain at a bit error rate (BER) of ${10}^{-5}$. Owing to the DC algorithm’s dual-layer decoding architecture, we can further improve the overall system performance through targeted optimization of either the code rate or memory size for communication sides with poor channel conditions, yielding an extra 0.2 dB gain at a BER of ${10}^{-5}$ compared to non-optimized configurations. The simulation results demonstrate that the proposed scheme significantly enhances maritime relay communication performance under harsh oceanic channel conditions while providing actionable insights for optimizing next-generation maritime communication system designs.

1. Introduction

With the widespread deployment of fifth-generation (5G) networks, terrestrial mobile communications have made unprecedented progress. However, the maritime domain presents unique challenges, including infrastructure deficiencies and complex electromagnetic propagation characteristics, making the direct transplantation of land-based communication solutions infeasible [1]. Non-terrestrial networks (NTNs) form an essential component of 5G-Advanced and sixth-generation (6G) systems, leveraging large-scale constellations to establish three-dimensional connectivity across the space, aerial, and maritime domains [2]. This multi-layered architecture specifically bridges oceanic coverage gaps while fulfilling 3GPP’s vision for ubiquitous service continuity [3]. Recently, as the research and development (R&D) of Space–Ground Integrated Networks (SGINs) is prioritized in many countries, maritime communication, as a critical application scenario, has attracted much attention in academia and industry [4]. A robust maritime communication network (MCN) is vital for diverse maritime operations, such as environmental monitoring, shipping logistics, and offshore activities within exclusive economic zones (EEZs) [5]. For example, accurate navigation data are a prerequisite for safe voyages, while real-time voice/video communication improves coordination during rescue missions. Additionally, infotainment services (e.g., web browsing and UHD video) are increasingly demanded by crew members and passengers. Therefore, establishing a reliable MCN is imperative to fully unlock the ocean’s potential for sustainable human development.

Currently, maritime communications mainly rely on coast/island-based cellular networks, maritime satellites, and legacy radio systems [6]. For cellular networks, the base stations (BSs) deployed along the coast can provide long-term evolution (LTE) or 5G services for maritime mobile terminals (MMTs), and their performance requirements can be reliably met. However, the transmission distance of the network is restricted by Earth’s curvature and the sea surface condition. At present, the maximum offshore coverage reported in the literature is limited to 100 km from the BS [7]. Maritime satellites are the main means of long-distance ship-to-ship/shore communication. Nevertheless, their transmission rates are low and delays are significant due to limited onboard payload capacity and long round-trip distances. Moreover, the costs of satellite launches, onboard equipment, and communication services are prohibitively high [8]. Maritime radio systems represented by automatic identification systems (AISs) operate in very high-frequency (VHF) bands. Although they support long-range communication, they cannot meet the required services for emerging maritime applications due to bandwidth limitations. Consequently, the existing maritime communication systems fail to satisfy the growing demands of offshore users.

To build a high-speed, cost-effective, and ultra-reliable MCN, numerous studies have been conducted. In [9,10,11,12], several maritime wireless channel measurement campaigns were reported, laying the foundation for robust MCN construction. By establishing a wireless mesh network among ships, buoys, and onshore BSs, the terrestrial network coverage can be extended to the sea [13]. In [14], tethered balloons acting as relay nodes (RNs) were deployed over the sea. The experiment results showed that the project could offer broadband services of 3 Mbps for users 150 km from the coast [15]. Similarly, leveraging their high altitude and agile maneuverability, unmanned aerial vehicles (UAVs) can deliver flexible communication services for offshore users in maritime environments [16,17]. The authors in [18] proposed a network based on distributed antennas (DAs) to meet the service requirements of MMTs and investigated the antenna selection scheme. Unlike the unpredictable movement of users on land, maritime users’ positions can be predicted more accurately as most ships follow established shipping lanes. In [19], a resource allocation scheme was proposed based on shipping lane information, which could improve the energy efficiency of the MCN. Despite these advancements, the development and optimization of channel coding techniques tailored for complex maritime environments remain a significantly understudied area in the literature.

To fill this gap, we focus on a class of coding techniques designed to overcome challenges in maritime environments. Specifically, in our prior work [20], we investigated the performance of block Markov superposition transmission (BMST), a class of spatially coupled codes that convert short codes into codes with long constraint lengths through time-domain Markov superposition [21]. This architecture enables BMST to mitigate burst errors caused by maritime multipath propagation while maintaining high encoding efficiency. Our simulations validated its robustness in ship-to-shore communication scenarios. Beyond such point-to-point links, we further investigate the application of BMST in maritime two-way relay networks (TWRNs) [22,23]. Such networks can provide beyond-line-of-sight (BLOS) connectivity for offshore terminals. However, conventional TWRNs enforce orthogonality by dividing the transmission resources (time/frequency) among users [24]. While orthogonal resource partitioning avoids interference, it inevitably results in spectral underutilization—a major bottleneck in maritime communication systems [25]. To concurrently address spectral efficiency, we employ physical-layer network coding (PNC), which converts symbol collisions into cooperative gains [26]. The synergistic integration of BMST and PNC provides a comprehensive solution that simultaneously tackles two critical challenges in maritime relay communications: (1) robust error correction against burst errors through BMST’s spatially coupled architecture and (2) enhanced spectral efficiency via PNC’s spatial-domain signal superposition. It should be noted that the initial simulations adopt an idealized assumption of equal path loss for both links in the TWRN, which simplifies the analysis but may underestimate performance variations in asymmetric channel conditions.

In this manuscript, we consider a more realistic maritime relay communication scenario and propose a more flexible BMST-coded PNC scheme. Due to space limitations, we focus solely on the decoding algorithm design of PNC to further enhance reliability, while another issue in PNC, i.e., asynchrony, is left for future work. Specifically, our main contributions include the following three aspects:

• We utilize log-normally distributed Rician factors to characterize maritime channel quality and rigorously evaluate the performance of the proposed scheme in TWRNs exhibiting asymmetric fading conditions across the two links.
• We introduce a novel co-design framework based on BMST and PNC, achieving simultaneous improvements in both bit error rate (BER) performance and spectral efficiency. Furthermore, we incorporate a class of fixed-length multi-rate codes into the BMST-PNC system to address heterogeneous channel conditions while allocating code rates through channel capacity analysis.
• We propose a novel iterative decoding and computing (DC) algorithm that sequentially executes iterative decoding prior to network coding operations. Benchmark evaluations demonstrate that the proposed DC algorithm achieves a 2.9 dB gain over a conventional computing and decoding (CD) algorithm at a BER of ${10}^{-5}$. Furthermore, through targeted unilateral optimization, the DC algorithm attains an additional 0.2 dB coding gain under asymmetric fading conditions, demonstrating remarkable robustness across heterogeneous channel environments.

The manuscript structure is as follows: Section 2 describes the system model and the maritime channel characteristics. Section 3 presents the encoding process and the DC algorithm for the BMST-coded PNC scheme. Section 4 provides a comprehensive analysis of the simulation results. Section 5 serves to conclude this manuscript. It summarizes the main findings and contributions of the research.

2. System Model and Channel Description

2.1. 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.

Figure 1. This is a space-ground integrated network framework.

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.

Figure 2. This is the signal processing procedure for UA phase of channel-coded PNC.

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 ${\mathit{u}}_i=\{u_{i,0},u_{i,1},\dots ,u_{i,2K-1}\}$ is encoded by a linear encoder, yielding a coded sequence ${\mathit{c}}_i=\{c_{i,0},c_{i,1},\dots ,c_{i,2N-1}\}$, where $i\in \{1,2\}$. We assume the coded sequence ${\mathit{c}}_i$ is then mapped to a symbol sequence ${\mathit{x}}_i=\{x_{i,0},x_{i,1},\dots ,x_{i,N-1}\}$ through the quadrature phase shift keying (QPSK) mapping, in which $x_{i,r}\in \mathcal{S}={\displaystyle \frac{1}{\sqrt{2}}}\{1+j,-1+j,1-j,-1-j\}$, $j=\sqrt{-1}$ and $0\le r\le N-1$. 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 $y_r\in \mathit{y}=\{y_0,y_1,\dots ,y_{N-1}\}$ can be calculated by

$$y_r=h_{1,r}{x}_{1,r}+h_{2,r}{x}_{2,r}+w_r,$$

(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 y, PNC mapper performs maximum-likelihood (ML) detection and exchanges soft information with two decoders iteratively. After iterative decoding, the estimate version $(\widehat{{\mathit{u}}_1},\widehat{{\mathit{u}}_2})$ of the original binary sequence pair $({\mathit{u}}_1,{\mathit{u}}_2)$ is obtained. The relay node then performs computing (network coding) operation, which can be viewed as a linear mapping from ${\mathbb{F}}_2^2$ to ${\mathbb{F}}_2$ [29]. Finally, the estimated network-coded sequence $\widehat{{\mathit{u}}_R}$ can be expressed as

$$\widehat{{\mathit{u}}_R}=\widehat{{\mathit{u}}_1}\oplus \widehat{{\mathit{u}}_2}.$$

(2)

Under ideal conditions, $\widehat{{\mathit{u}}_R}$ is equivalent to ${\mathit{u}}_R={\mathit{u}}_1\oplus {\mathit{u}}_2$.

During the DB phase, the relay node $N_R$ re-encodes the obtained sequence ${\mathit{u}}_R$ and broadcasts it to both source nodes. Knowing its own information ${\mathit{u}}_1$, the source node $N_1$ can extract the binary sequence ${\mathit{u}}_2$ transmitted by node $N_2$ from the network-coded sequence ${\mathit{u}}_R$. This process can be described as an XOR operation, i.e., ${\mathit{u}}_1\oplus {\mathit{u}}_R={\mathit{u}}_2$. Similarly, the source node $N_2$ can also extract the binary sequence transmitted by node $N_1$. Compared to the complex processing of the UA phase at the relay node, involving techniques such as mapping and modulo-2 summation, the issue during the DB phase resembles common point-to-point transmissions and thus requires no special treatment [30]. Therefore, we focus solely on the BER performance of node $N_R$ during the UA phase, and the details regarding the terminal receivers are omitted.

2.2. Maritime Wireless Channels

Compared with terrestrial communications, marine communications possess a distinctive and prominent characteristic, namely sparsity, which is crucial to determining the performance and operational modalities of marine communication systems. This sparsity is primarily manifested in two significant aspects: one is sparse scattering, and the other is sparse user distribution [31]. Since there are few obstacles over the sea, the scattering of electromagnetic waves is sparse. Consequently, line-of-sight (LoS) paths prevail in most maritime communication links. From this perspective, the Rayleigh fading model, which is commonly adopted in terrestrial communications with rich scattering, is not suitable for marine communications. The inadequacy of Rayleigh fading stems from its assumption of omnidirectional scattering, which contradicts the sparse scattering nature of maritime environments. In contrast, the Rician model’s explicit incorporation of a dominant LoS component aligns with empirical observations of over-sea propagation, where non-line-of-sight (NLoS) paths are typically limited to sea surface bounce and sporadic atmospheric refraction.

As stated in [9], the Rician fading model, consisting of LoS and NLoS components, is capable of predicting the propagation behavior of electromagnetic waves over the sea. The channel coefficient h under Rician fading can be modeled as

$$h=\sqrt{{\displaystyle \frac{\mathcal{K}}{\mathcal{K}+1}}}+\sqrt{{\displaystyle \frac{1}{\mathcal{K}+1}}}{h}_s,$$

(3)

where $\mathcal{K}$ is known as Rician factor. It quantifies the relative strength of LoS versus NLoS components [32]. A high $\mathcal{K}$ value indicates that the LoS component dominates, suggesting a relatively clear and less obstructed path for the electromagnetic waves, while a low $\mathcal{K}$ value implies a more complicated environment with significant contributions from the NLoS component due to multiple reflections and scatterings. Here, $h_s$ is modeled as a complex Gaussian random variable with zero mean and unit variance. Notably, when $\mathcal{K}=0$, Rician fading is equivalent to Rayleigh fading. In [10], a log-normal distribution with mean ${\mu }_{\mathcal{K}}$ and variance ${\sigma }_{\mathcal{K}}^2$ is proposed to characterize the Rician factor $\mathcal{K}$ at sea. This means that the signal attenuation at sea exhibits exponential scaling in dB-domain measurements. The $\mathcal{K}$ satisfies the equation

$$\tilde{\mathcal{K}}=10·{log}_{10}\mathcal{K}\sim \mathcal{N}({\mu }_{\mathcal{K}},{\sigma }_{\mathcal{K}}^2).$$

(4)

The log-normal distribution of $\mathcal{K}$ reflects the composite effects of maritime channel variations, including sea surface roughness, atmospheric ducting, and evaporation layer effects. Throughout this manuscript, we use ${\mu }_{\mathcal{K},i}$ and ${\sigma }_{\mathcal{K},i}$ to denote the mean and the standard deviation of Rician factor characterizing the channel between source node $N_i$ and relay node $N_R$, respectively, $i\in \{1,2\}$.

3. Physical-Layer Network Coding Based on Block Markov Superposition Transmission

3.1. Encoding Algorithm

To mitigate asymmetric channel impairments in maritime TWRNs, we employ the repetition and single-parity-check (RSPC) codes as the basic codes for BMST. These fixed-length multi-rate codes, constructed via time-sharing between the repetition (R) codes and the single-parity-check (SPC) codes, inherently satisfy PNC’s fundamental requirement of equal-length signal blocks from source nodes while accommodating link asymmetry. Let $\mathcal{C}[n,k]$ denote a binary linear code with an input of k bits and an output of n bits. For any given integer $n>1$, the R code and the SPC code can be denoted as $\mathcal{C}[n,1]$ and $\mathcal{C}[n,n-1]$, respectively. Mathematically, the structure of an RSPC code is given by the Cartesian product of ${\mathcal{C}}^{\alpha }[n,1]\times {\mathcal{C}}^{\beta }[n,n-1]$. This implies that the R code is utilized $\alpha$ times in the procedure, while the SPC code is utilized $\beta$ times. The structure of an RSPC code is shown in Figure 3, and the code rate $\mathcal{R}$ can be expressed as

$$\mathcal{R}={\displaystyle \frac{\alpha }{\alpha +\beta }}·{\displaystyle \frac{1}{n}}+{\displaystyle \frac{\beta }{\alpha +\beta }}·{\displaystyle \frac{n-1}{n}}.$$

(5)

For a target code rate ranging from $1/n$ to $(n-1)/n$, we can implement it by setting $\beta =k-1$ and $\alpha =n-2-\beta$. It is evident that the RSPC codes with varying code rates possess the same length $(\alpha +\beta )n=n(n-2)$, which indicates that they can be encoded/decoded without hardware reconfiguration through adjustments in ($\alpha ,\beta$). Taking $n=4$ and $k=2$ as an example, the generator matrix G corresponding to the RSPC code has the following form:

$$\mathit{G}=\left[\begin{array}{cccccccc}1& 1& 1& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 1\\ 0& 0& 0& 0& 0& 1& 0& 1\\ 0& 0& 0& 0& 0& 0& 1& 1\end{array}\right].$$

(6)

For more details, refer to [33].

To boost the reliability of maritime data transfer, BMST coding is applied at the source node. During this process, each codeword is replicated and then superimposed onto subsequent codewords for transmission (see Figure 4 for reference). This approach achieves coding gain through repetition while guaranteeing transmission efficiency. BMST enables the upgrade construction from arbitrary short/weak basic codes to long/strong codes through its unique encoding mechanism. The encoding procedure for source node $N_i$ is shown in Figure 5, in which $\boxed{\mathrm{D}}$, $\boxed{\mathsf{\Pi }}$, and ⨁ represent a shift register, an interleaver, and the modulo-2 addition, respectively.

Let ${\mathit{u}}_i^{\left(t\right)}=(u_{i,0}^{\left(t\right)},u_{i,1}^{\left(t\right)},\cdots ,u_{i,k(n-2)-1}^{\left(t\right)})\in {\mathbb{F}}_2^{k(n-2)}$ be the t-th block of the binary sequence generated by source node $N_i$. The binary sequence ${\mathit{u}}_i=({\mathit{u}}_i^{\left(0\right)},{\mathit{u}}_i^{\left(1\right)},\cdots ,{\mathit{u}}_i^{(L-1)})$ with L blocks is sent to the encoder of the RSPC code $\mathcal{C}\left[n\right(n-2),k(n-2\left)\right]$ in turn, yielding a coded sequence ${\mathit{c}}_i=({\mathit{c}}_i^{\left(0\right)},{\mathit{c}}_i^{\left(1\right)},\cdots ,{\mathit{c}}_i^{(L+m-1)})$ with $L+m$ blocks, where m, called the memory, is the number of registers in the BMST system. Each sub-block ${\mathit{c}}_i^{\left(t\right)}=(c_{i,0}^{\left(t\right)},c_{i,1}^{\left(t\right)},\cdots ,c_{i,n(n-2)-1}^{\left(t\right)})\in {\mathbb{F}}_2^{n(n-2)}$, $0\le t\le L+m-1$ can be expressed by

$${\mathit{c}}_i^{\left(t\right)}={\mathit{u}}_i^{\left(t\right)}\mathit{G}{\mathbf{\Pi }}_0+{\mathit{u}}_i^{(t-1)}\mathit{G}{\mathbf{\Pi }}_1+˯+{\mathit{u}}_i^{(t-m)}\mathit{G}{\mathbf{\Pi }}_m,$$

(7)

where $\mathit{G}$ is a $k(n-2)\times n(n-2)$ generator matrix of RSPC code and ${\mathbf{\Pi }}_j(0\le j\le m)$ is a $n(n-2)\times n(n-2)$ permutation matrix. Note that, for $t<0$, ${\mathit{u}}_i^{\left(t\right)}$ is set to a zero vector initially. The detailed encoding algorithm is presented in Algorithm 1.

Algorithm 1: Encoding algorithm for node $N_i$, $i\in \{1,2\}$
Initialization:
Set ${\mathit{v}}_i^{\left(t\right)}=\mathbf{0}\in {\mathbb{F}}_2^{n(n-2)}$ for all $t<0$.
for $t=0$ to $L+m-1$ do
		if $t<L$ then
				Split ${\mathit{u}}_i^{\left(t\right)}\in {\mathbb{F}}_2^{k(n-2)}$ into:
				Left-most $\alpha$ bits → Encode into $\alpha$ R codewords;   Remaining bits → Encode into $\beta$ SPC codewords;
				Combine to form an RSPC codeword ${\mathit{v}}_i^{\left(t\right)}\in {\mathbb{F}}_2^{n(n-2)}$;
		else
				Set ${\mathit{u}}_i^{\left(t\right)}=\mathbf{0}$ and ${\mathit{v}}_i^{\left(t\right)}=\mathbf{0}$; // Termination
		end
		for $j=0$ to m do
				Calculate ${\mathit{w}}_i^{(t,j)}={\mathit{v}}_i^{(t\mathsf{-}j)}{\mathsf{\Pi }}_j$;
		end
		Calculate the t-th block ${\mathit{c}}_i^{\left(t\right)}={\sum }_{0\le j\le m}{\mathit{w}}_i^{(t,j)}$.
end

Because of the termination operation, the effective code rate ${\mathcal{R}}^{\prime }$ of BMST-RSPC is reduced to ${\displaystyle \frac{L}{L+m}}$ of the basic code rate $\mathcal{R}$. Notably, when L is sufficiently large (i.e., $L\gg m$), the system’s code rate asymptotically approaches $\mathcal{R}$, rendering the rate loss negligible. Therefore, there is no need to worry about the code rate loss. Additionally, since the interleaving (from ${\mathit{v}}_i$ to ${\mathit{w}}_i$) and the encoding (from ${\mathit{u}}_i$ to ${\mathit{v}}_i$) are executed in parallel, the overall encoding latency of the BMST-RSPC system can be comparable to that of the basic code, provided that parallel processing resources are available.

3.2. Iterative Decoding and Computing Algorithm

In PNC, the CD algorithm maps received superimposed symbols into network-coded signals through a non-bijective mapping process—multiple distinct signal combinations may be mapped to identical network-coded symbols. As critically noted in [34], this inherent many-to-one characteristic of PNC mapping during the uplink phase inevitably leads to loss of network-coding-related information embedded in the received symbols. To overcome this fundamental limitation, we propose the DC algorithm, which performs the computational operation after iterative decoding between two decoders. As shown in Figure 6, a normal graph is utilized to represent the BMST-PNC system, in which edges and vertices denote variables and constraints, respectively.

Each edge carries a message representing the probability mass function (PMF) of its associated variable [35]. The vertex $\boxed{\mathrm{C}}$ represents the constraint of RSPC code. The vertex $\boxed{=}$ enforces equality constraints on connected variables, whereas the vertex $\boxed{+}$ imposes a linear constraint requiring the sum of connected variables to be zero in the binary field. The vertex $\boxed{\phi }$ computes the so-called extrinsic messages, sent to one decoder, based on the likelihood functions and the feedback messages from the other decoder. The dual-layer structure exhibits strong flexibility, allowing for code rate and memory adjustment provided that the transmission symbol lengths from both source nodes remain equal.

Throughout this paper, QPSK modulation is applied, that is, an RSPC codeword with $n(n-2)$ bits are mapped to ${\displaystyle \frac{n(n-2)}{2}}$ symbols. For brevity, we use ${\mathit{c}}_r^{\left(t\right)}=({\mathit{c}}_{1,r}^{\left(t\right)},{\mathit{c}}_{2,r}^{\left(t\right)})=(c_{1,2r}^{\left(t\right)},c_{1,2r+1}^{\left(t\right)},c_{2,2r}^{\left(t\right)},c_{2,2r+1}^{\left(t\right)})\in {\mathbb{F}}_2^4$ to represent the coded sequence corresponding to the symbol pair $(x_{1,r}^{\left(t\right)},x_{2,r}^{\left(t\right)})$, $0\le r<{\displaystyle \frac{n(n-2)}{2}}$. Upon receiving the r-th superposition symbol $y_r^{\left(t\right)}$, the likelihood function corresponding to the symbol pair $(x_{1,r}^{\left(t\right)},x_{2,r}^{\left(t\right)})$ can be computed by

$$f\left(y_r^{\left(t\right)}|x_{1,r}^{\left(t\right)},x_{2,r}^{\left(t\right)},h_{1,r}^{\left(t\right)},h_{2,r}^{\left(t\right)}\right)\propto \exp \left(-{\displaystyle \frac{{\left|y_r^{\left(t\right)}-h_{1,r}^{\left(t\right)}{x}_{1,r}^{\left(t\right)}-h_{2,r}^{\left(t\right)}{x}_{2,r}^{\left(t\right)}\right|}^2}{2{\sigma }^2}}\right),$$

(8)

where $x_{i,r}^{\left(t\right)}\in \mathcal{S}$, and $h_{i,r}^{\left(t\right)}$ represents the time-varying channel response from source node $N_i$ to relay node $N_R$, $i\in \{1,2\}$. For the sake of description, we use $P_{{\mathit{c}}_r^{\left(t\right)}}^{(+\to \phi )}\left(z\right)$ and $P_{{\mathit{c}}_r^{\left(t\right)}}^{(\phi \to +)}\left(z\right)$, $z\in {\mathbb{F}}_2$, to denote the a priori message and the extrinsic message via ${\mathit{c}}_r^{\left(t\right)}$, respectively. The message update algorithm for PNC mapper is detailed in Algorithm 2.    

Algorithm 2: Message update algorithm

Input: Provide $f\left(y_r^{\left(t\right)}|x_{1,r}^{\left(t\right)},x_{2,r}^{\left(t\right)},h_{1,r}^{\left(t\right)},h_{2,r}^{\left(t\right)}\right)$ and the a priori messages $P_{c_{r,l}^{\left(t\right)}}^{(+\to \phi )}\left(z\right)$ as inputs, where $z\in {\mathbb{F}}_2$    and $0\le l\le 3$; Output: For $0\le l\le 3$, output the extrinsic messages calculated by $$P_{c_{r,l}^{\left(t\right)}}^{(\phi \to +)}\left(z\right)\propto \sum _{c_{r,l}^{\left(t\right)}=z}f\left(y_r^{\left(t\right)}|x_{1,r}^{\left(t\right)},x_{2,r}^{\left(t\right)},h_{1,r}^{\left(t\right)},h_{2,r}^{\left(t\right)}\right)\prod _{\begin{array}{c}j=0,\\ j\ne l\end{array}}^3{P}_{c_{r,j}^{\left(t\right)}}^{(+\to \phi )}\left(c_{r,j}^{\left(t\right)}\right).$$

The BMST-RSPC decoder employs an iterative sliding-window decoding algorithm with delay d. 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 $\boxed{=}$ of degree $m+2$, $n(n-2)$ parallel nodes $\boxed{+}$ of degree $m+2$, and a node of type $\boxed{\mathrm{C}}$. 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→Decoder 1→PNC mapper→Decoder 2→PNC mapper→⋯→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.   

Algorithm 3: The decoding process for BMST-RSPC system at Time t
Input:
Extrinsic messages $P_{{\mathit{c}}_r^{\left(t^{\prime }\right)}}^{(\phi \to +)}\left(z\right)$ for $t-d\le t^{\prime }\le t$, $z\in {\mathbb{F}}_2$;   Maximum iteration number $J_{max}>0$;   Entropy threshold ${\delta }_h>0$;
Iteration:
for $J=1$ to $J_{max}$ do
		for $t^{\prime }=t-d$ to t do
				Process messages through vertices in order: $\boxed{+}\to \boxed{\mathsf{\Pi }}\to \boxed{=}\to \boxed{\mathrm{C}}\to \boxed{=}\to \boxed{\mathsf{\Pi }}\to \boxed{+}$;
				// Forward Pass
		end
		for $t^{\prime }=t$ to $t-d$ do
				Process messages through vertices in order: $\boxed{+}\to \boxed{\mathsf{\Pi }}\to \boxed{=}\to \boxed{\mathrm{C}}\to \boxed{=}\to \boxed{\mathsf{\Pi }}\to \boxed{+}$;
				// Backward Pass
		end
		Termination Check:
		Compute hard decision estimation ${\widehat{{\mathit{u}}_i}}^{(t-d)}$;   Estimate the entropy rate $h_J\left({\mathit{y}}^{(t-d)}\right)$;   If $\left|h_J\left({\mathit{y}}^{(t-d)}\right)-h_{J-1}\left({\mathit{y}}^{(t-d)}\right)\right|\le {\delta }_h$, then exit the loop;
end
Output:
Estimated bits ${\widehat{{\mathit{u}}_i}}^{(t-d)}$;   Updated messages $P_{{\mathit{c}}_r^{\left(t^{\prime }\right)}}^{(+\to \phi )}\left(z\right)$ for $t-d\le t^{\prime }\le t$.

Algorithm 4: Iterative decoding and computing algorithm for BMST-RSPC-PNC system
Initialization: For $0\le t\le d-1$,
Initialize all messages over the edges within and connecting to the t-th   decoding layer of both decoders using uniform distribution;   Compute the likelihood function according to (8);   Set the maximum iteration count between two decoders $I_g>0$.
Iterative decoding:
for $t=d$ to $L+d-1$ do
		if $t\le L+m-1$ then
				Initialize messages over edges within/connecting to the t-th decoding layer using uniform distribution and compute the likelihood function according to (8);
		end
		for $I=1$ to $I_g$ do
				for $t^{\prime }=t-d$ to$min(t,L+m-1)$do
						PNC Mapper: Update extrinsic messages from $\boxed{\phi }$ to $\boxed{+}$ via ${\mathit{c}}_1^{\left(t^{\prime }\right)}$ using Algorithm 2, and forward messages to Decoder 1;
						Decoder 1: Execute sliding-window decoding per Algorithm 3, and then feed back messages from $\boxed{+}$ to $\boxed{\phi }$ via ${\mathit{c}}_1^{\left(t^{\prime }\right)}$ to PNC mapper;
						PNC Mapper: Update extrinsic messages from $\boxed{\phi }$ to $\boxed{+}$ via ${\mathit{c}}_2^{\left(t^{\prime }\right)}$ using Algorithm 2, and forward messages to Decoder 2;
						Decoder 2: Execute sliding-window decoding per Algorithm 3, and then feed back messages from $\boxed{+}$ to $\boxed{\phi }$ via ${\mathit{c}}_2^{\left(t^{\prime }\right)}$ to PNC mapper.
				end
		end
		Computing: Calculate the estimated network-coded packet according to (2).
end

3.3. Capacity Analysis

As we all know, channel capacity, usually defined as the maximum achievable mutual information, represents the ultimate limit on the information rate for reliable communication over a channel. Shannon 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 $({\mathit{x}}_1,{\mathit{x}}_2)$ and the received superposition signal y, conditioned on the channel gains $({\mathit{h}}_1,{\mathit{h}}_2)$, can be expressed as

$$\begin{array}{cc}\hfill I({\mathit{x}}_1,{\mathit{x}}_2;\mathit{y}|{\mathit{h}}_1,{\mathit{h}}_2)& =I({\mathit{x}}_1;\mathit{y}|{\mathit{h}}_1,{\mathit{h}}_2)+I({\mathit{x}}_2;\mathit{y}|{\mathit{x}}_1,{\mathit{h}}_1,{\mathit{h}}_2)\hfill \\ \hfill & =I({\mathit{x}}_2;\mathit{y}|{\mathit{h}}_1,{\mathit{h}}_2)+I({\mathit{x}}_1;\mathit{y}|{\mathit{x}}_2,{\mathit{h}}_1,{\mathit{h}}_2),\hfill \end{array}$$

(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 ${\mathit{x}}_1$ and the received superposition variable y can be decomposed as

$$\begin{array}{cc}\hfill I({\mathit{x}}_1;\mathit{y}|{\mathit{h}}_1,{\mathit{h}}_2)& =H\left({\mathit{x}}_1\right)-H\left({\mathit{x}}_1\right|\mathit{y},{\mathit{h}}_1,{\mathit{h}}_2)\hfill \\ & ={log}_2\left(\right|\mathcal{S}\left|\right)+\mathbb{E}[{log}_{2}Pr\left\{{\mathit{x}}_1\right|\mathit{y},{\mathit{h}}_1,{\mathit{h}}_2\}],\hfill \end{array}$$

(10)

and

$$\begin{array}{cc}\hfill I({\mathit{x}}_1;\mathit{y}|{\mathit{x}}_2,{\mathit{h}}_1,{\mathit{h}}_2)& =H\left({\mathit{x}}_1\right)-H\left({\mathit{x}}_1\right|\mathit{y},{\mathit{x}}_2,{\mathit{h}}_1,{\mathit{h}}_2)\hfill \\ & ={log}_2\left(\right|\mathcal{S}\left|\right)+\mathbb{E}[{log}_{2}Pr\left\{{\mathit{x}}_1\right|\mathit{y},{\mathit{x}}_2,{\mathit{h}}_1,{\mathit{h}}_2\}].\hfill \end{array}$$

(11)

Here, $H(·)$ denotes the entropy, and $\left|\mathcal{S}\right|$ represents the number of constellation points in $\mathcal{S}$, respectively. The $\mathbb{E}[·]$ 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. Thus, the channel capacity $C_1$ between node $N_1$ and relay node $N_R$ is given by

$$C_1={\displaystyle \frac{I({\mathit{x}}_1;\mathit{y}|{\mathit{h}}_1,{\mathit{h}}_2)+I({\mathit{x}}_1;\mathit{y}|{\mathit{x}}_2,{\mathit{h}}_1,{\mathit{h}}_2)}{2}}.$$

(12)

Similarly, the channel capacity $C_2$ between node $N_2$ and relay node $N_R$ is given by

$$C_2={\displaystyle \frac{I({\mathit{x}}_2;\mathit{y}|{\mathit{h}}_1,{\mathit{h}}_2)+I({\mathit{x}}_2;\mathit{y}|{\mathit{x}}_1,{\mathit{h}}_1,{\mathit{h}}_2)}{2}}.$$

(13)

3.4. Genie-Aided Lower Bound Analysis

BMST codes exhibit a notable advantage in that their performance can be effectively characterized through genie-aided lower bounds, which are obtained using a genie-aided decoder. The basic idea behind the genie-aided decoder is stated as follows:

For any given $\mathit{\ell }\ge 0$, we assume the existence of a genie that provides the decoder with complete information about all layers except the ℓ-th decoding layer. In other words, the decoder has perfect knowledge of the partial information sequences ${\underline{\mathit{u}}}^{\prime }=({\mathit{u}}^{\left(0\right)},\cdots ,{\mathit{u}}^{(\mathit{\ell }-1)},{\mathit{u}}^{(\mathit{\ell }+1)},\cdots ,{\mathit{u}}^{(L-1)})$. Under this assumption, it follows that the BER performance of the standard BMST decoder cannot surpass that of the genie-aided decoder. Consequently, the performance of the proposed DC algorithm is lower-bounded by the iterative decoding process between two genie-aided decoders. In our simulations, the genie-aided decoder is implemented by setting $L=1$, which simplifies the analysis while preserving the essential characteristics of the bound. Theoretical derivations and performance descriptions of the genie-aided decoder are detailed in [21].

4. Simulation Results and Discussion

In this part, several numerical examples are given to illustrate the performance of the BMST-RSPC-PNC system. These examples demonstrate the capabilities of the proposed scheme. In all the simulations, QPSK modulation is employed, and the channel is modeled as a fast-fading Rician channel, where ${\mu }_{\mathcal{K},1}=10.0$ dB, ${\mu }_{\mathcal{K},2}=3.0$ dB, and ${\sigma }_{\mathcal{K},1}={\sigma }_{\mathcal{K},2}=0.0$ dB. Given that the combined transmission power from the two senders and the noise variance is $2{\sigma }^2$, the signal-to-noise ratio (SNR) is equal to ${\displaystyle \frac{1}{{\sigma }^2}}$. The decoding delay d is fixed at 20. The parameters in Algorithm 3 are set as $J_{max}=18$ and ${\delta }_h={10}^{-5}$. The detailed simulation configurations are presented in

Table 1. Simulation parameters.

Parameter	Value
The mean ${\mu }_{\mathcal{K},1}$	10.0 dB
The standard deviation ${\sigma }_{\mathcal{K},1}$	0.0 dB
The mean ${\mu }_{\mathcal{K},2}$	3.0 dB
The standard deviation ${\sigma }_{\mathcal{K},2}$	0.0 dB
The decoding delay d	20
Maximum iteration number $J_{max}$	18
Entropy threshold ${\delta }_h$	${10}^{-5}$

.

4.1. The Impact of Iteration Count on Performance

The number of iterations $I_g$ in the iterative decoding between the two BMST decoders significantly impacts system performance. We construct an RSPC code $\mathcal{C}{[20,1]}^9\times \mathcal{C}{[20,19]}^9$ with a code rate of $\mathcal{R}=0.5$ (denoted as $\mathcal{C}[360,180]$) and take the 10-fold Cartesian product $\mathcal{C}{[360,180]}^{10}$ as the basic code for both BMST encoders.

Figure 7. This includes the BER curves of information sequences ${\mathit{u}}_R$, ${\mathit{u}}_1$, and ${\mathit{u}}_2$ under different iteration numbers. Each encoder of the BMST-RSPC-PNC system terminates every 200 data sub-blocks, i.e., $L_1=L_2=200$ and $m_1=m_2=5$. The code rate of the BMST-RSPC is ${\mathcal{R}}^{\prime }=0.4878$.

Figure 7. This includes the BER curves of information sequences ${\mathit{u}}_R$, ${\mathit{u}}_1$, and ${\mathit{u}}_2$ under different iteration numbers. Each encoder of the BMST-RSPC-PNC system terminates every 200 data sub-blocks, i.e., $L_1=L_2=200$ and $m_1=m_2=5$. The code rate of the BMST-RSPC is ${\mathcal{R}}^{\prime }=0.4878$.

Figure 7 shows the BER curves of information sequences under varying iteration counts. We find that, as the iteration count increases, the performance gains of the information sequences saturate, particularly beyond four iterations. It is worth noting that, when SNR = 6.4 dB, increasing the iteration count $I_g$ from 4 to 5 yields negligible BER improvement across all sequences. This indicates that, under high-SNR conditions, the decoding process may reach a saturation point where more iterations do not significantly improve performance but still waste computational resources. Additionally, the BER of the network-coded sequence ${\mathit{u}}_R$ is always slightly higher than that of the information sequence ${\mathit{u}}_2$, which is transmitted over the poorer channel. This confirms that the weaker channel in the marine TWRN degrades relay performance.

4.2. The Impact of Different Algorithms on System Performance

In this subsection, $I_g$ is set to 4, and the basic codes for BMST encoders are the same as those used in Section 4.1. The BER performance of the BMST-RSPC-PNC system using different algorithms is shown in Figure 8.

For the BMST-RSPC-PNC system with $m_1=m_2=5$, compared with the CD algorithm, the DC algorithm can obtain a coding gain of 2.9 dB at the BER of ${10}^{-5}$. A similar phenomenon can be observed in the BMST-RSPC-PNC system with $m_1=m_2=6$. This improvement arises because the DC algorithm performs computations after separate decoding, effectively avoiding the loss of useful information. In low-SNR regimes, the elevated BER is primarily attributed to error propagation during the iterative decoding process, where residual errors in the initial stages progressively accumulate and amplify across subsequent iterations. However, in the high-SNR region, the effect of error propagation is not observed. We can also find that the proposed system has comparable waterfall performance with different memories and that the error floor can be further lowered by increasing the memory. The proposed system maintains stable waterfall performance over different memory configurations while achieving progressively lower error floors with increased memory. For the DC algorithm, the BER performance of the system at 7 dB can be lowered from ${10}^{-5}$ to ${10}^{-6}$ by simply increasing the memory from 5 to 6. The experimental results demonstrate close alignment between all the performance metrics and their corresponding lower bounds in high-SNR conditions.

4.3. The Impact of Different Optimization Approaches on System Performance

In this example, we explore the probability of further improving the system performance under the dual-layer decoding architecture. The mutual information versus SNR for the PNC scheme with QPSK modulation is shown in Figure 9. The Shannon capacity limit for the relay node to attain 1.9 bits/channel-use is observed (node $N_1$ corresponds to 1.0 bits/channel-use and node $N_2$ corresponds to 0.9 bits/channel-use) to be 5 dB. Consequently, node $N_1$ and node $N_2$ achieve code rates of 0.5 and 0.45, respectively. Based on the above analysis, we construct an RSPC code $\mathcal{C}{[20,1]}^{10}\times \mathcal{C}{[20,19]}^8$ with a code rate of $\mathcal{R}=0.45$ (marked as $\mathcal{C}[360,162]$).

As shown in Figure 10, we plot the BER curve of the BMST-RSPC-PNC system with $m_1=m_2=6$ and ${\mathcal{R}}_1={\mathcal{R}}_2=0.5$ (marked as SM-SR) as a benchmark. We find that error propagation significantly degrades system performance in low-SNR regimes, while its impact gradually diminishes in high-SNR regions.

For the scheme with the same memory and different code rates ($m_1=m_2=6$, ${\mathcal{R}}_1=0.5$, and ${\mathcal{R}}_2=0.45$), marked as SM-DR, we take $\mathcal{C}{[360,180]}^{10}$ and $\mathcal{C}{[360,162]}^{10}$ as the basic codes for BMST encoders at node $N_1$ and node $N_2$, respectively. For the $L_1=L_2=200$ setup, the code rates ${\mathcal{R}}_1^{\prime }$ and ${\mathcal{R}}_2^{\prime }$ are 0.4854 and 0.4369, respectively. For the scheme with different memories and the same code rate ($m_1=6$, $m_2=7$, and ${\mathcal{R}}_1={\mathcal{R}}_2=0.5$), marked as DM-SR, the basic codes for both BMST encoders are $\mathcal{C}{[360,180]}^{10}$. To keep the number of decoding layers the same, we set $L_1=200$ and $L_2=199$. Hence, the code rates ${\mathcal{R}}_1^{\prime }$ and ${\mathcal{R}}_2^{\prime }$ are 0.4854 and 0.4830, respectively. Compared with the SM-SR scheme, the SM-DR scheme has a more significant performance improvement in the waterfall region. We also find that, when the SNR exceeds 6.5 dB, the DM-SR scheme has a lower error floor. This is because increasing the memory (the repeated transmission numbers) of the BMST system can lower its error floor, hence mitigating the impact of the inferior-performance side on the overall performance of the system. The performance evaluation shows that, at the target BER of ${10}^{-5}$, both the SM-DR and DM-SR schemes yield an extra 0.2 dB coding gain.

5. Conclusions

This study investigates a practical maritime TWRN scenario and proposes an efficient and reliable data transmission scheme that combines BMST codes with PNC. By leveraging superposition operations in both the time and spatial domains, the proposed design simultaneously enhances reliability and spectral efficiency, addressing a key challenge in maritime communications. To mitigate information loss at the relay node, we introduce a novel DC algorithm that performs network coding operations after iterative decoding. The numerical results demonstrate that, under identical parameter configurations, the BMST-RSPC-PNC system employing the DC algorithm achieves a remarkable 2.9 dB coding gain over the CD algorithm at a BER of ${10}^{-5}$. Capitalizing on the flexible dual-layer decoding architecture of the DC algorithm, we further propose two enhanced schemes, i.e., the SM-DR scheme and the DM-SR scheme, to optimize relay node performance. The theoretical and empirical analyses reveal that the SM-DR scheme exhibits superior waterfall performance, while the DM-SR scheme attains a lower error floor, rendering them adaptable to diverse operational requirements in complex maritime environments.

The findings of this work provide valuable insights for developing ultra-reliable and high-efficiency maritime communication systems. Within the framework of the SGIN, the proposed scheme presents a promising solution for long-distance maritime transmission, with potential applications in next-generation wireless networks.