# BER Minimization by User Pairing in Downlink NOMA Using Laser Chaos Decision-Maker
## 5 Parameter Optimization for NOMA System with Laser Chaos Decision-Maker and Error Correcting Codes
### Performance Evaluation with ECC  
We demonstrate the simulation results regarding the dependency on ECC and other parameters related to laser chaos decision-maker.  
First, we evaluate the BER with different \(ECC_{\varphi,n}\). In this simulation, \(ECC_{\varphi,n}\) is configured as either \(ECC_{0,n}\), \(ECC_{1,n}\), \(ECC_{3,n}\), or \(ECC_{5,n}\). In addition, parameter \(Z\), related to the laser chaos decision-maker, is set to one. The cell radius is varied from 100 to 1000. Figure 9a shows the simulation results. From Figure 9a, we observe BER becomes smaller as the cell radius is smaller. Furthermore, it should be noted that the optimal \(ECC_{\varphi,n}\) differs depending on the cell radius. The setting \(ECC_{0,n}\) performs the best BER when the cell radius is smaller than about 450 m, whereas the setting \(ECC_{0,n}\) cannot not perform the best BER when the cell radius is larger than 450 m. That is, here we confirm that an appropriate setting of \(ECC_{\varphi,n}\) does matter.  
Next, we evaluate the BER with adaptive \(ECC_{\varphi,n}\), where \(\varphi\) is set as an adaptive value. The value of \(\varphi\) is defined as the average bit errors of the last \(\tau\) times for user \(n\). Adaptive \(ECC_{\varphi,n}\) at step \(t\) can be expressed as: \(\varphi=\sum_{t-1}^{t-\tau}\frac{bit\text{-errors}}{\tau}\). We denote \(ECC_{\tau 5,n}\) as the adaptive \(ECC_{\varphi,n}\) with \(\tau=5\), meaning that the ECC is the average bit errors of the last five transmission times for user \(n\). In this simulation, the parameter \(Z\) for the laser chaos decision-maker is set to one. Figure 9b summarizes the simulation results while changing the cell radius from 100 m to 1000 m. From Figure 9b, with a larger cell radius, we can see that the adaptive ECC method with \(\tau=7\) accomplishes smaller BER in the NOMA system while \(\tau=7\) can get smaller BER than other settings. With a smaller cell radius, on the other hand, the setting of \(ECC_{0,n}\) accomplishes the smallest BER.  
We also evaluate the effect of the number of thresholds on BER of the proposed LCDM-NOMA. Figure 10 shows simulation results. While maintaining \(\varphi\) being adaptive with \(\tau\) value of 7, \(Z\) is configured differently by 1, 2, 4, 8, 16, 32, 64, and 128. Meanwhile, in the case of \(ECC_{0,n}\), \(ECC_{1,n}\), \(ECC_{3,n}\), and \(ECC_{5,n}\), \(Z\) is set to 1. Remember that the number of thresholds is given by \(2Z+1\). When the number of available threshold levels is small, the threshold can reach the upper or lower limit more quickly. Thus, the convergence of the selection becomes generally fast, whereas accurate selection becomes more difficult.  
Figure 9: BER with different \(ECC_{\varphi,n}\) settings; (**a**) BER comparison with a fixed ECC. (**b**) BER comparison with an adaptive ECC.  
Conversely, with a larger number of thresholds, it becomes more likely that the threshold reaches its upper or lower limit through sufficient exploration. Hence, the convergence needs a longer time duration but may lead to the correct selection. Figure 10b,c are the enlargement of Figure 10a when the cell radius varies from 100 m to 500 m and from 500 m to 1000 m, respectively. From Figure 10b regarding a smaller cell radius, the BER becomes smaller as the number of thresholds increases. Meanwhile, from Figure 10c with respect to a larger cell radius, the BER becomes smaller as the number of threshold steps is reduced. Therefore, we can conclude that appropriate setting of the number of thresholds improves the system performance depending on the given cell radius. Specifically, \(Z=4\) is the best when the cell radii are 1000 m and 900 m. \(Z=8\) is the best when the cell radii are 800 m and 700 m. \(Z=\)8 and \(Z=64\) are the best when the cell radii are 600 m and 500 m, respectively. \(Z=128\) and \(Z=64\) are the best when the cell radii are 400m and less than 350 m, respectively.