# 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
### Parameter Optimization  
Section 5.2 demonstrated that a proper parameter settings of \(ECC_{\varphi,n}\) and \(Z\) indeed works concerning the cell radius in improving the BER performances. Based on these observations, here we introduce another parameter denoted by \(r_{n}\) that can take three values instead of the former binary (ACK or NACK) to improve the BER performance. When the \(n\)th user experiences zero bit errors, it sends \(r_{n}=2\) to the BS. When the number of bit errors is larger than 0 but not larger than \(\varphi_{n}\), it sends \(r_{n}=1\) to the BS. Finally, when the number of bit errors is larger than \(\varphi_{n}\), \(r_{n}=0\) is transmitted to the BS. Namely, \(r_{n}\) is expressed as follows:  
\[r_{n}= \begin{cases}2,&\text{if }\text{{bit-error}}_{n}=0.\\ 1,&\text{if }\text{0}<\text{{bit-error}}_{n}\leq\varphi_{n}.\\ 0,&\text{if }\text{{bit-error}}_{n}>\varphi_{n}.\end{cases}  \tag{10}\]  
On the BS, ACK or NACK affect the reward as defined before. Since \(r_{n}\) is set to three values in this case, we define that the selected pairing option wins (i.e., gets a reward) if \(r_{n}\geq 1\) holds for all \(n\). Otherwise, the selected pairing option loses (i.e., gets no reward). Figure 10: **(a)** BER comparison with the different number of thresholds \(2Z+1\) related to the laser chaos decision-maker. (ECC is set to \(ECC_{\tau 7}\) for \(Z=1,\dots,128\). \(Z\) is set to 1 for \(ECC_{0,n}\), \(ECC_{1,n}\), \(ECC_{3,n}\), \(ECC_{5,n}\)). **(b)** Magnified view of **(a)** with respect to smaller cell radii; **(c)** magnified view of **(a)** with respect to smaller cell radii. We evaluate the performance in terms of BER of the proposed LCDM-NOMA considering parameter optimization via \(r_{n}\), and compare it with the case without considering parameter optimization. The solid line in Figure 11 demonstrates the BER when the parameter optimization is employed in LCDM-NOMA, whereas other lines represent the cases without parameter optimization. In the optimization, ECC is also dynamically reconfigured. ECC is given by \(ECC_{\tau 7,n}\) for all users; namely, ECC is adaptively configured with \(\tau=7\). Meanwhile, \(Z\) is set, as the best \(Z\) corresponds to the setting of the cell radius before \(\sum_{n=1}^{N}r_{n}=2N\) is satisfied. Once \(\sum_{n=1}^{N}r_{n}=2N\) is achieved, the ECC is set as \(ECC_{0,n}\) while \(Z\) is set as 1. This setting means that ECC is adaptively reconfigured when bit errors are present, whereas once the error-free situation is accomplished, ECC does not look back at past information. From Figure 11, the optimized LCDM-NOMA exhibits lower BER than other cases thanks to the three-valued \(r_{n}\) and dynamic ECC setting reconfiguration. Finally, the optimized LCDM-NOMA is compared with C-NOMA, UCDG-NOMA and an \(\epsilon\)-greedy-based user-pairing scheme. Herein, \(l_{v}\) and \(s_{v}\) are defined as the number of selected \(v\)th pairing options and the number of successful communications, respectively. The communication success-rate for the \(v\)th pairing \(p_{v}\) can be expressed as \(p_{v}=s_{v}/l_{v}\). In the \(\epsilon\)-greedy-based user pairing scheme, the pair option is selected randomly with probability \(\epsilon\), and the pairing option with the highest communication success rate with probability \(\epsilon\), and the pairing option which has the highest value of \(p_{v}\) with probability \(1-\epsilon\) is selected. We evaluate two cases of \(\epsilon\) by 0.1 and 0.01. From Figure 12, it is remarkable that the optimized LCDM-NOMA exhibits the smallest BER when the cell radius is small compared with other strategies. The reason that our proposed scheme is better than the others can be summarized as follows. C-NOMA and UCGD-NOMA are fixed pairing schemes, which pair users based on their location information without considering BER. On the other hand, our proposed LCDM-NOMA can pair the users based on ACK information that depends on BER by updating the threshold scheme according to the real communication process.