# Ciphered BCH Codes for PAPR Reduction in the OFDM in Underwater Acoustic Channels
## 5 Simulation Setup and Results
### PAPR Performance
For a 64-subcarrier OFDM system, the maximum gain obtained was approximately 10 dB for a ciphered \(\text{BCH}(63,7)\) OFDM system, which means that the maximum PAPR of the proposed system never exceeded 8.06 dB. Similarly, for a 128-subcarrier OFDM system, the maximum gain obtained was approximately 12 dB for a ciphered \(\text{BCH}(127,8)\) OFDM system. Figure 6: Possible PAPR gain from the most appropriate key obtained from a set of multiple random keys. Figure 8 compares the PAPR performances of the proposed 32-subcarrier OFDM systems with uncoded and BCH-encoded OFDM systems using CCDF plots. Figure 8a,c,e compare the PAPR performances of ciphered \(\mathrm{BCH}(31,k)\) with those of uncoded systems, while Figure 8b,d,f compare the PAPR performances of ciphered \(\mathrm{BCH}(31,k)\) with those of \(\mathrm{BCH}(31,k)\) encoded systems. For the sake of illustration, Figure 8a shows the gain achieved in PAPR as an example for a 32-subcarrier system. As shown in Figure 7, the PAPR performance of the ciphered \(\mathrm{BCH}(31,6)\) system was much superior to that of an uncoded OFDM system, as the PAPR of proposed system never exceeded about 7.5 dB. As shown in Figure 8b, when comparing the PAPR performance of a ciphered \(\mathrm{BCH}(31,6)\) with that of conventional \(\mathrm{BCH}(31,6)\) system, it was observed that the PAPR of a conventional \(\mathrm{BCH}(31,6)\)-encoded system remained high with a high probability, unlike the uncoded OFDM system. A similar phenomenon is observed in Figure 8d. As would be expected, the PAPR performance of ciphered \(\mathrm{BCH}\) codes for higher values of \(k\) was not at par with the systems where the value of \(k\) was small, as shown in Figure 8e,f. This is because when the code rate \(k/n\) becomes large, there are fewer number of possible key vectors that will produce low PAPR sequences. Hence, there does not exist a key vector that would give the same gain in PAPR as is given by the key vector when the code rate is low. Figure 7: Gain in maximum PAPR as a function of various ciphered \(\mathrm{BCH}\) codes. Figure 8: PAPR performance of the proposed technique for ciphered BCH(31,\(\lambda\)) systems. (**a**) Ciphered BCH(31,6) vs uncoded OFDM, (**b**) Ciphered BCH(31,6) vs original BCH(31,6), (**c**) Ciphered BCH(31,11) vs uncoded OFDM, (**d**) Ciphered BCH(31,11) vs original BCH(31,11), (**e**) Ciphered BCH(31,16) vs uncoded OFDM and (**f**) Ciphered BCH(31,16) vs original BCH(31,16). However, it must be noted that although there is not an appreciable gain in PAPR for the higher-rate ciphered BCH codes, their PAPR performances are still much better than the conventional BCH codes. As a result, the bit error rate performance is significantly improved without any deterioration in PAPR performance. Figure 9 compares the PAPR performances of the proposed 64-subcarrier OFDM systems with uncoded and BCH-encoded OFDM systems using CCDF plots. Figure 9a,c,e compare the PAPR performances of ciphered \(\mathrm{BCH}(63,k)\) systems with those of uncoded systems, while Figure 9b,d,f compare the PAPR performances of ciphered \(\mathrm{BCH}(63,k)\) systems with those of \(\mathrm{BCH}(63,k)\) encoded systems. As shown in Figure 9a, the PAPR performance of the ciphered \(\mathrm{BCH}(31,6)\) system was much superior to that of an uncoded OFDM system as it never exceeded about 8 dB, while that of uncoded scheme increased to about 13 dB. As shown in Figure 9b, when comparing the PAPR performance of the ciphered \(\mathrm{BCH}(63,10)\) with that of the conventional \(\mathrm{BCH}(63,10)\) system, it was observed that the PAPR of the conventional \(\mathrm{BCH}(63,10)\) encoded system remained high with a high probability, unlike the uncoded OFDM system--a phenomenon that was also observed in the case of the 32-subcarrier system. A similar phenomenon is observed in Figure 9d. In this case also, the PAPR performance of the ciphered BCH codes for higher values of \(k\) was not at par with the systems where the value of \(k\) was small, as shown in Figure 9e,h.