## 4 Simulation Results
As the SNR increases, the performance gap between the two schemes gradually decreases, and, at an SNR of 10.75 dB, the BERs of both relay schemes become nearly identical. This suggests that, in high SNR conditions, the two schemes can achieve comparable telecommunication performance. In comparison to the PNC-Polar relay scheme, the DNC-Polar relay scheme exhibits superior performance in terms of BER. This is because, in the PNC-Polar scheme, the relay node relies on superimposed signals, which are affected by noise, to compute soft information, potentially leading to inaccuracies. Consequently, errors may arise during multiple decoding operations, degrading the quality of bidirectional communication. In contrast, the DNC-Polar relay scheme ensures that signals from source nodes are processed independently, eliminating the potential for interference caused by signal superposition. Each signal is decoded individually, resulting in fewer errors and enhancing the reliability of the VLC system. In terms of transmission efficiency and cost-effectiveness, the PNC-Polar relay scheme outperforms the DNC-Polar relay scheme. Specifically, it necessitates only two time slots and a single set of receiving equipment at the relay node, as opposed to the three time  
Figure 8:0 The BER performance of different relay schemes in a VLC system. slots and two sets of equipment required by the DNC-Polar scheme. This advantage of the PNC-Polar scheme significantly enhances the VLC system's transmission efficiency and minimizes equipment costs. Furthermore, at high SNR levels, the PNC-Polar relay scheme demonstrates comparable telecommunication performance with reduced operational complexity. The simulated BER outcomes validate the practicality and effectiveness of this innovative scheme. Figure 17 shows the BER results of the uncoded case and the joint design scheme of polar code and PNC in the VLC relay system. In these simulations, the code length is set to 1024, and the code rates are 0.5, 0.7, and 0.9. As shown in the results, the BER curve of the joint design scheme is significantly lower than that of the uncoded case. The proposed scheme effectively corrects erroneous data generated in telecommunication and improves the reliability of visible light signals. At a BER of \(10^{-2}\), a coding gain of approximately 3 dB is observed between the uncoded case and the 0.5 code rate condition. This performance gain is expected to increase with higher SNR levels. At the same time, the relay node adopts the specific mapping to process the superimposed signals directly, converting the mutual interference between them into valuable information that aids in the completion of telecommunications. This leads to a reduction in transmission time slots and an increase in network throughput capacity. The proposed scheme's efficacy and feasibility are thus confirmed. Figure 18 illustrates the BER results of the VLC relay system, adopting the proposed scheme with varying code lengths. The experiments cover code lengths ranging from 128 to 4096, all at a fixed code rate of 0.5. Generally, as the code length increases, a more pronounced downward trend in the BER curve is observed. When the BER exceeds \(10^{-2}\), the performance distinctions among different code lengths are relatively minor. However, for BERs lower than \(10^{-2}\), code length variations have a notable impact on the error control capability of the VLC system. Therefore, in real-world communication scenarios, it is crucial to consider actual requirements and select an appropriate code length. It is worth noting that excessively long codes may lead to increased decoding latency, potentially affecting communication timeliness. Figure 19 shows the effective bit rates per unit time-slot of the VLC system that adopts different relay schemes. The effective bit rates per unit time-slot of the three relay  
Figure 8:1 The BER performance of the proposed scheme across different code rates. schemes gradually increase with the growth of SNR.