# Ciphered BCH Codes for PAPR Reduction in the OFDM in Underwater Acoustic Channels
## 4 Proposed Technique  
We propose a ciphered BCH (C-BCH)-encoded OFDM scheme that uses predetermined keys to obtain the lowest possible PAPR for an \(N\) sub-carrier system. It may be noted that the proposed key selection algorithm is offline and several keys that would produce low PAPR are determined for each code. These key vectors are then used in the live system to produce ciphered codes for reducing the PAPR, as depicted in Figure 5, improving the error performance and providing secure communication. Since the key vectors are obtained in advance using a computer, real time calculations for determining low PAPR combinations are not needed at the transmitter. The only overhead is an XOR operation; however, this operation also adds a level of encryption to the system. The \(n\) value of the BCH(\(n,k\)) encoder is selected based on the number of bits required in one OFDM symbol. For example, for an OFDM system with 64 subcarriers and binary phase-shift keying (BPSK) modulation, BCH(\(63,k\)) would be selected. BCH(\(63,k\)) would also be used for an OFDM system that has 32 subcarriers and quadrature phase-shift keying (QPSK) modulation since in this case, one OFDM symbol will have 64 bits as well. The value of \(k\) is based on the desired error and PAPR performances. It is known that the lower the code rate \((k/n)\), the better the error performance. Additionally, such lower code rates can also translate to higher PAPR gains using the proposed algorithm--as will be shown in Section 5. Hence, a BCH(\(63,10\)) code will give a better error performance as well as better PAPR gain than a BCH(\(63,45\)) code. Algorithm 1 summarizes the steps for determining the best key vectors for an \(N\) sub-carrier BPSK-OFDM scheme and works as follows. A BCH(\(n,k\)) encoder is chosen based upon the value of \(N\), such that \(n=N-1\). Since the value of \(k\) is a design parameter, it is chosen based on the desired error performance and required PAPR gain. If the value of \(k\) is reasonably small, such that all possible \(k\)-bit words can be generated, then \(N_{R}\), i.e., the number of \(k\)-bit words to be generated, is given by \(2^{k}\). If the value of \(k\) is large enough, such that it is computationally impractical to generate all possible \(k\)-bit words, then  
One key vector is chosen from the set of \(N_{w}\) key vectors and all the BCH-encoded words that were generated in the previous step are XORed with this key vector, resulting in a new set of BCH codes called \"ciphered BCH codes\". The PAPR of each of these ciphered codes is computed and the maximum value is stored. Then, the next key vector is chosen from the set of \(N_{w}\) key vectors and the process of producing ciphered codes, determining their PAPRs and storing the maximum PAPR is repeated until all \(N_{w}\) key vectors are exhausted. This results in \(N_{w}\) values of maximum PAPRs along with their corresponding key vectors. Finally, the minimum value from the stored maximum PAPRs is selected along with its corresponding key vector. Since there will be multiple instances of minimum PAPRs and their corresponding key vectors, several key vectors are selected for later use in the live system. The number of key vectors to be selected is a design parameter that depends upon the desired level of security. The higher the level of desired security, the more the number of key vectors selected. The selected key vectors are then used in the live system in a round-robin fashion to perform the XOR operation as shown in Figure 2. The set of selected key vectors are shared with the receiver before switching the system to live operation. However, in the proposed system, side information is not required to be transmitted to the receiver. A single bit (either 0 or 1) is appended to each of the encoded symbols to make them \(N\)-bits long. This step is necessary for two reasons. First, higher PAPR gains are possible only when the number of bits in the encoded sequence are equal to the number of bits in one OFDM symbol.