## 2 Preliminaries
### Relay Schemes  
Wireless communication systems are prone to mutual interference, which occurs when electromagnetic waves operating on nearby frequencies disrupt each other. This interference can ultimately degrade both accuracy and overall performance [23]. When signals share the same channel, one signal can act as interference for another. This is the reason why, in the conventional relaying scheme, only one node is authorized to transmit data during a specific time slot [24]. This is depicted in the schematic diagram in Figure 2a:  
* In the first time slot, Node A transmits a signal \(x_{A}\) to the relay node;
* During the second time slot, the relay node receives the signal \(x_{A}\) from Node A and then relays it to Node B;
* In the third time slot, Node B sends a signal \(x_{B}\) to the relay node;
* Finally, in the fourth time slot, the relay node forwards the signal \(x_{B}\) that it received from Node B to Node A.  
To enable an exchange of information between A and B, four time slots are required, leading to an information transmission efficiency of \(1/4\).  
Shown in Figure 2b is a schematic diagram illustrating the Direct Network Coding (DNC) relay scheme:  
* During the first time slot, Node A transmits a signal \(x_{A}\) to the relay node;
* In the second time slot, Node B transmits a separate signal \(x_{B}\) to the relay node;
* In the third time slot, the relay node applies an XOR operation to the two received signals (\(x_{A}\oplus x_{B}\)) and broadcasts the resultant signal to both A and B simultaneously.  
Figure 2: Comparison of different relay schemes. (**a**) The traditional relay scheme. (**b**). The relay scheme employs direct network coding (DNC). (**c**) The model of the physical-layer network coding (PNC) relay scheme.  
To avoid mutual interference, A and B transmit their signals in separate time slots. Once receiving the signals, the relay node employs DNC, often in the form of an XOR operation, as an encoding process [25]. The relay node then simultaneously broadcasts the encoded results to both A and B. Subsequently, node A retrieves the desired information by performing an XOR operation between the received signal and its original signal, resulting in \(x_{A}\oplus x_{B}\oplus x_{A}=x_{B}\). Node B follows an identical procedure. This approach requires a total of 3 time slots for A and B to successfully exchange information, resulting in an information transmission efficiency of 1/3. In comparison to the conventional relay scheme, this approach enhances the VLC system's throughput by 33%.  
In contrast to the aforementioned relay schemes, which treat simultaneous channel signals as interference to be avoided, the PNC relay mode transforms interference into useful information, thereby enhancing the performance of the VLC system. The schematic diagram of the PNC relay scheme is illustrated in Figure 2c:  
* During the first time slot, A and B transmit their respective signals, \(x_{A}\) and \(x_{B}\), to the relay node simultaneously;
* During the second time slot, the relay node processes the combined signal by applying a predefined mapping method, which effectively encodes the signals. The relay then simultaneously broadcasts this encoded signal to both A and B.  
The PNC relay scheme reduces the time required by one slot, resulting in a 50% increase in throughput compared to the DNC relay scheme. It also saves two time slots, effectively doubling the throughput when compared to the traditional relay scheme. In summary, the PNC relay scheme considerably enhances the throughput.