## 4 Comparative Analysis Convolutional Codes with LDPC Codes, BCH Codes, and Turbo Codes
### Convolutional Codes vs. BCH Codes  
BCH codes are a class of cyclic error-correcting codes that are constructed using polynomials over a finite field. They have precise control over the number of symbol errors correctable by the code and can be decoded easily using an algebraic method known as syndrome decoding. They are used in various applications such as satellite communications, DVDs, QR codes, and quantum-resistant cryptography. One way to compare the complexity of BCH codes and Convolutional codes is to look at their encoding and decoding algorithms. In order to explicate the intricacies involved, we shall consider a (15, 7) BCH code and a (2, 1, 2) Convolutional code as illustrative examples. These codes have been chosen due to their comparable error-correcting capabilities. This is solely a simplified illustration, and the actual complexity may vary depending on the implementation and the code parameters. BCH codes: They are a class of cyclic error-correcting codes that are constructed using polynomials over a finite field. They have precise control over the number of symbol errors correctable by the code and can be decoded easily using an algebraic method known as syndrome decoding. They are used in various applications such as satellite communications, DVDs, QR codes, and quantum-resistant cryptography. BCH encoding: It is the process of finding a polynomial that is a multiple of the generator polynomial, which is defined as the least common multiple of the minimal polynomials of some consecutive powers of a primitive element. The encoding can be either systematic or non-systematic, depending on how the message is embedded in the codeword polynomial. To encode a message using BCH code, we need to choose a finite field GF(q), a code length n, and a design distance d. Then, we need to find a generator polynomial g(x) that has \(\propto\), \(\propto^{2}\),..., \(\propto^{\text{(d-1)}}\) as roots, where \(\propto\) is a primitive element of GF(q\({}^{\text{m}}\)) and n = q\({}^{\text{m}}-1\). The table of minimal polynomials in reference [18] can be used to find g(x) as the least common multiple of some of them. Then, multiply the message polynomial m(x) by g(x) to obtain the codeword polynomial c(x). BCH decoding: This is the process of finding and correcting the errors in the received codeword using the syndromes, which are formed by evaluating the received polynomial at some powers of a primitive element. There are several algorithms for decoding BCH codes, such as [PERSON]'s algorithm, the Forney algorithm, [PERSON]'s algorithm, and the Euclidean algorithm. They involve finding the error locator polynomial and the error values, and then subtracting them from the received codeword to recover the original one. To decode a received word r(x) using BCH code, we need to calculate the syndromes \(s_{j}=r(\alpha^{d})\) for j = 1, 2,..., 2t, where t is the number of errors that can be corrected. Then, use an algorithm, such as [PERSON]'s algorithm or the Euclidean algorithm, to find the error locator polynomial \(\wedge(x)\) that has the error locations as roots. Then, use another algorithm, such as [PERSON]'s algorithm, to find the error values at those locations. Finally, correct the errors by subtracting them from the received word to get the original codeword. For example, suppose we want to encode the message [1 0 1 0] using a binary BCH code with n = 15 and d = 7, it is recommended to choose GF(2) as the finite field and use the reducing polynomial \(z^{4}+z+1\) and the primitive element \(\propto(z)=z\) as in reference [18]. Then, we can find the generator polynomial g(x) as follows:  
\[g(x)=\text{lcm}(m_{1}(x),m_{2}(x),\dots,m_{6}(x))=(x^{4}+x+1)(x^{4}+x^{3}+x^{2}+ x+1)=x^{8}+x^{7}+x^{6}+x^{4}+1\]  
Please note that the lcm, which stands for least common multiple, is responsible for ensuring that g(x) contains all the necessary roots required for BCH encoding.