# Enhancing BER Performance Limit of BCH and RS Codes Using Multipath Diversity +
## 7 Results  
In this paper, analysis of BCH and RS codes over AWGN and Rayleigh channels has been conducted. The block length of (\(n_{1}\), \(k_{11}\)) and (\(n_{1}\), \(k_{12}\)) for the BCH code where \(k_{11}>k_{12}\), and (\(n_{2}\), \(k_{21}\)) and for RS code where \(k_{21}>k_{22}\) and (\(n_{2}\), \(k_{22}\)) are simulated with three, five, and seven paths. The threshold value was set at 40-75% from the SNR of the LoS signal to choose the NLoS signals. The  
Figure 4: BER vs. SNR of CWMC, when three, five, seven, and nine paths are combined. (**a**) The AWGN channel; and (**b**) the Rayleigh channel. Figure 5: BER vs. SNR of CWMC with different threshold values and channels. (**a**,**b**) The threshold value is 80% from the LoS signal SNR. (**c**,**d**) The threshold value is 20% from the LoS signal SNR. BER ratio was computed by changing E\({}_{b}\)/N\({}_{0}\) from 1 to 25. In the simulation results, the red slope represents the uncoded signal (LoS). The black and the black with stars slopes represent the LoS which is encoded by using codes where \(t=1\), and \(t=2\), respectively. The blue slope represents the combining of the coded LoS with two coded NLoS, where LoS and NLoS are encoded by using codes where \(t=1\). Figures 6 and 7 show that the BCH and RS codes in a multipath transmission consistently perform better than the BCH and RS codes without CWMC in AWGN and Rayleigh channels under a binary environment. It can be seen that combining three, five, and seven coded paths result in a better slope than solely the coded LoS. The absolute BER performance of BCH and RS codes is improved by approximately 3 dB and between 8 and 9 dB over AWGN and Rayleigh channels, respectively, for three combining paths. The combining of five paths improved the BER performance by approximately 4 dBm and from 11 to 12 dB over AWGN and Rayleigh channels. The combining of seven paths improved the performance of BER more than the combining of three or five paths. The improvement was 6 dB over the AWGN channel and between 13 and 14 dB over the Rayleigh channel at a BER of \(10^{-3}\). BCH (15,11), BCH (127,120), and BCH (255,247) codes have an error correction capability \(t_{1}=1\), while the error correction capability of BCH (15,7), BCH (127,113), and BCH (255,239) codes is \(t_{2}=2\). Figures 8-10, show that the combining of three paths improved the performance of BCH (15,11), BCH (127,120), and BCH (255,247) codes. The performance is improved by approximately 3 dB, 2.8 dB, and  
Figure 6: BER vs. SNR of the BCH (15,11) code with and without CWMC. (**a**) The AWGN channel; and (**b**) the Rayleigh channel. Figure 7: BER vs. SNR of the RS (15,13) code with and without CWMC. (**a**) Over the AWGN channel; and (**b**) over the Rayleigh channel. 2.75 dB, respectively, over the AWGN channel, and 7 dB, 10 dB, and 11.5 dB, respectively, over the Rayleigh channel at a BER of \(10^{-3}\). Furthermore, their performance is better than BCH (15,7), BCH (127,113), and BCH (255,239) codes by approximately 0.75 dB, 0.85 dB, and 1 dB, respectively, over the AWGN channel, and 2 dB, 6.2 dB, and 7.5 dB, respectively, over the Rayleigh channel at a BER of \(10^{-3}\). Similarly, the RS (15,13), RS (  
Figure 11: BER vs. SNR of the RS (15,13) code with and without CWMC vs. the RS (15,11) code without CWMC, where 3P represents the three paths that were combined to improve the error correction capability of the RS (15,13) code. (**a**) The AWGN channel; and (**b**) the Rayleigh channel. Figure 12: BER vs. SNR of the RS (127,125) code with and without CWMC vs. the RS (127,123) code without CWMC, where 3P represents the three paths that were combined to improve the error correction capability of the RS (127,125) code. (**a**) The AWGN channel; and (**b**) the Rayleigh channel. Figure 10: BER vs. SNR of the BCH (255,247) code with and without CWMC vs. the BCH (255,239) code without CWMC, where 3P represents the three paths that were combined to improve the error correction capability of the BCH (255,247) code.