# Ciphered BCH Codes for PAPR Reduction in the OFDM in Underwater Acoustic Channels
## 4 Proposed Technique
Second, to facilitate the efficient computation of the FFT algorithm, the number of sub-carriers in an OFDM system are kept in powers of two. There are three options for appending the extra bit to the ciphered BCH code. One option is to add this bit arbitrarily. This will make the total encoded bits equal to \(N\) and will save computational power. However, the added bit will not be used and will be wasted. The second option is to utilize this bit for channel estimation. This may marginally increase the computational complexity of the system; however, it can improve the error performance of the system. The third option is to use this bit to obtain further gains in PAPR. However, to determine  
Figure 5: An example of the generation of ciphered BCH(n,\(k\)) codes for an 8 subcarrier OFDM system with \(k=4,\ n=7\). whether appending a bit 0 or a bit 1 provides more gains in PAPR, the PAPR will have to be computed twice for each OFDM symbol: once while appending a binary 1 and again while appending a binary 0. Our observation is that the computational power required in the third option far outweighs the gain obtained in PAPR. ```
1. Initialize \(N_{w},\ N_{R}\), \(j=0\)
2. Repeat\(N_{R}\) times
3. Generate a \(k\)-bit binary sequence \(R_{j}\)
4. Encode \(R_{j}\) using \(\text{BCH}(n,k)\) and generate encoded sequence \(E_{j}\)
5. \(j=j+1\)
6. \(i=0\)
7. Repeat\(N_{w}\) times
8. Randomly generate an \(n\)-bit key vector \(k_{i}\)
9. \(j=0\)
10. Repeat\(N_{R}\) times
11. \(C_{j}=k_{i}\oplus E_{j}\)
12. \(\text{PAPR}_{j}=\text{PAPR}\Big{\{}C_{j}\Big{\}}\)
13. \(j=j+1\)
14. \(\text{PAPR}^{i}_{max}=\underset{\forall j}{\max}\text{PAPR}_{j}\)
15. \(i=i+1\)
16. \(\text{PAPR}_{min}=\underset{\forall i}{\min}\text{PAPR}^{i}_{max}\)
17. \(\text{\emph{Key}}_{best}=k_{l}\)
```  
**Algorithm 1** Key Vector Selection  
Hence, out of the three options, the second option is the most feasible. Table 2 lists some of the key vectors computed for several BCH codes using the proposed algorithm. For each BCH code, we use four key vectors and use them in rotation after every subcarrier block. The vectors and their rotation orders are known at the receiver side; thus, the data is encrypted and only a receiver with the knowledge of the key vectors used will be able to decipher the received information. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Code**} & **No. of** & **No. of Vectors** & **Key Vectors (Hex)** &
\begin{tabular}{c} **Gain over** \\ **BCH (dB)** \\ \end{tabular} \\ \hline \multirow{4}{*}{BCH (31,6)} & & 900 \(\times\) 10\({}^{3}\) & 0 DA1 BDB0 & 7.6588 \\  & All 2\({}^{k}\) & 10 \(\times\) 10\({}^{3}\) & 5813A4C0 & 7.4783 \\  & & 50 \(\times\) 10\({}^{3}\) & 7C2157D4 & 7.4397 \\  & & 100 \(\times\) 10\({}^{3}\) & 243A2D840 & 7.6109 \\  & & 30 \(\times\) 10\({}^{3}\) & 0C912 BDC & 4.9975 \\ BCH (31,11) & All 2\({}^{k}\) & 30 \(\times\) 10\({}^{3}\) & 31 CF7963 & 4.9975 \\  & & 30 \(\times\) 10\({}^{3}\) & 45 EE06B4 & 4.9975 \\  & & 30 \(\times\) 10\({}^{3}\) & 038C0550 & 4.9975 \\  & & 10 \(\times\) 10\({}^{3}\) & 778D144A4 & 3.2545 \\ BCH (31,16) & All 2\({}^{k}\) & 10 \(\times\) 10\({}^{3}\) & 6E9 EE556 & 3.2545 \\  & & 2500 & 37D46C5D0 & 3.2545 \\  & & 2500 & 4599D8C7 & 3.2545 \\  & & 20 \(\times\) 10\({}^{3}\) & 31 DCO4A527297604 & 9.8923 \\ BCH (63,7) & All 2\({}^{k}\) & 50 \(\times\) 10\({}^{3}\) & 64F1 EA51 AFD4 CF7E & 9.7317 \\  & & 50 \(\times\) 10\({}^{3}\) & 344 DDDD2044A4 BEAC & 9.8782 \\  & & 50 \(\times\) 10\({}^{3}\) & 292410 ED70648C3 & 9.8218 \\  & & 10 \(\times\) 10\({}^{3}\) & 3 AAP8436466653B & 8.5194 \\ BCH (63,10) & All 2\({}^{k}\) & 10 \(\times\) 10\({}^{3}\) & 25 FD2809 AB0312185 & 8.5194 \\  & & 20 \(\times\) 10\({}^{3}\) & 571232 AF4372046 & 8.5194 \\  & & 10 \(\times\) 10\({}^{3}\) & 17 AC03 EF085814D & 8.5194 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Keys selected and gains achieved.