# Ciphered BCH Codes for PAPR Reduction in the OFDM in Underwater Acoustic Channels
## 5 Simulation Setup and Results
### PAPR Performance  
In this work, \"gain in PAPR\" refers to reduction in maximum possible PAPR, and the maximum possible PAPR for an OFDM system with \(N\) number of subcarriers is given by \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Quantity** \\ \hline TX-RX distance & 500, 1500 m \\ Depth & 10 m \\ Max doppler shift & 10 Hz \\ Gain vector & [0; \(-1.5;-2.5;-2]\) dB \\ Tau vector & [0; \(1;2.5;7]\) ms \\ Atmospheric pressure & \(1.01325\times 10^{5}\) Pa \\ Salinity & 35 parts/1000 \\ Density & \(10^{3}\) Kg/m\({}^{3}\) \\ Water temperature & \(25\,\mathrm{\SIUnitSymbolCelsels}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Channel parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Code**} & \multicolumn{2}{c}{**No. of**} & \multirow{2}{*}{**No. of Vectors**} & \multirow{2}{*}{**Key Vectors (Hex)**} & **Gain over** \\  & & & & **BCH (dB)** \\ \hline \multirow{8}{*}{BCH (63,16)} & \multirow{8}{*}{All 2\({}^{\mathrm{k}}\)} & 1000 & 4 CEFF656704A38285 & 6.0206 \\  & & 1000 & 6E892F8204C475 FE & 6.0206 \\  & & 1000 & 4 CE8A16F57087981 & 6.5812 \\  & & 1000 & 4 CE28C0 DD58608D & 6.0206 \\  & & 6000 & 11 BCEB858D97 ASD & 1.4927 \\  & & 500 & 42B4 DF5 FA1F308A0 & 1.5664 \\  & & 1000 & 3D4 DD2C765C0 AEC & 1.0067 \\  & & 400 & 3380 FB6 BC55 EF3 & 1.4927 \\  & & 100 & 3938 BB20C10407 FA82B26D73D5D6 AC & 11.6499 \\  & & 100 & 2E5678670 CC3D50F32840268A11 AB4E & 11.7250 \\  & & 500 & 3D4303A31A73A437A3523 FB0F45C4853 & 11.5146 \\  & & 100 & 125 FA632718151512857367159C525 & 11.7058 \\  & & 300 & 455 EA09475B1341A41C375 FA59A56 FB08108 & 7.8241 \\  & & 1000 & 1E27792B1A3 EE2E60890738 DCDC4854B & 8.5194 \\  & & 1000 & 6E2029980 BCD2049399B65763785B & 8.5194 \\  & & 100 & 7C2721205129B2E25C777 EC20C3 & 8.1648 \\  & & 100 & 51874 DF0E5A3505 DEFA7C12 EADDE812E & 1.8035 \\  & & 372 ACD8B3D4781 FB35A581939A6821 & 1.7430 \\  & & 100 & 0B90 BF572C00420837 AAAAAAAA68370 & 1.5278 \\  & & 100 & 12D3A52A5943884 FB9D71181A722 & 1.8035 \\  & & 100 & 6E9372 CS186A95C4 BE47 BQ5F435 ADB & 0.9485 \\  & & 100 & 0E695015B32B2A58D4809A5 CA1 FA5 CA5A0 & 0.6437 \\  & & 100 & 7 FA4C71E8A54F9C9411B9893B1055 CE6 & 0.9485 \\  & & 100 & 3C13 DC9529792E6 EC58 AE77 AB3C8D0D & 0.8899 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(Cont.\) \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Quantity** \\ \hline No. of subcarriers & 32, 64, 128 \\ Mapping & BPSK \\ FFT size & 32, 64, 128 \\ CP length & N/4 \\ Equalizer & Zero Forcing \\ Bandwidth & 10 kHz \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Transceiver parameters. \(10\)\(\log_{10}\)\(N\). We simulated three OFDM systems with 32, 64 and 128 subcarriers. Ciphered \(\text{BCH}(31,\ k)\) codes and the associated keys were used for the 32-subcarrier system, ciphered \(\text{BCH}(63,\ k)\) and the associated keys for the 64-subcarrier system and ciphered \(\text{BCH}(127,\ k)\) and the associated keys for the 128-subcarrier system. Although, for large values of \(k\), a large number of possible keys are randomly generated, it was observed that using a very large pool of randomly generated keys did not produce a key that would give any appreciable gain in PAPR. In other words, a small subset of randomly generated keys is enough to provide the most appropriate keys that provide a substantial gain in PAPR. One such example is shown in Figure 6, where increasing the pool of random keys only provides a marginal improvement in PAPR, and a 6 dB gain is possible using a key obtained from a set of 50 randomly generated keys. Figure 7 shows the gain in maximum PAPR for the three types of ciphered BCH codes used in this work. It should be noted that the smaller the value of \(k\), the greater the gain in PAPR. For a 32-subcarrier OFDM system, the maximum gain obtained was approximately 8 dB for a ciphered BCH(31,6) OFDM system. The maximum possible PAPR for an uncoded 32-subcarrier OFDM system was 15.05 dB; this means that the maximum PAPR of the proposed system never exceeded 7.05 dB.