## 4 Comparative Analysis Convolutional Codes with LDPC Codes, BCH Codes, and Turbo Codes
### Convolutional Codes vs. LDPC Codes  
LDPC codes are known for their efficiency; however, decoding LDPC codes is more challenging compared to decoding Convolutional codes. Furthermore, LDPC codes require a larger memory capacity to store the parity check matrix used in the decoding process than Convolutional codes. To comprehend the reasons behind the higher memory requirements for LDPC codes in comparison to Convolutional codes, we can observe the following:  
In LDPC codes [22; 23], the code matrix, denoted as the parity matrix \(\mathbf{H}\), plays a vital role in establishing the relationship between the message and the codeword. The parity matrix consists of (N-K) rows and N columns, where N represents the block length and K represents the message length. This matrix is essential for both encoding and decodingprocedures, as it guides the transformation of messages into codewords and vice versa. The memory required to store the code matrix is directly proportional to the number of elements in the matrix, which is (N-K) \(\times\) N.  
The decoding process for LDPC codes involves the utilization of the message-passing algorithm. This decoding method employs the parity matrix **H** and specific rules to update the probabilities of each bit in the codeword. The iterations of these updates continue until a solution is reached or a maximum limit is attained. The memory needed for decoding the code using the message-passing algorithm is proportional to the number of elements in the matrix **H**, multiplied by the number of iterations \(I\), which yields ((N-K) \(\times\) N) \(\cdot\) I. In Convolutional codes [10; 24], the generator matrix **G** is a (_k,n_) matrix consisting of polynomial entries. The generator matrix **G** defines the combination of input bits and shift register bits to generate the output bits. Each row in **G** corresponds to one output bit, and each column represents one input bit or one shift register bit. For example, if \(\textbf{G}= \begin{pmatrix}1&1&0\\ 0&1&1\end{pmatrix} \), it implies that the encoder has \(k\) = 2 output bits, \(n\) = 1 input bit, and \(m\) = 2 shift register bits, and this is because the generator matrix for a convolutional code is a matrix that describes how the input bits are encoded into output bits by using shift registers and modulo-2 additions. The number of rows of the generator matrix is equal to the number of output bits per input bit, which is denoted by \(k\). The number of columns of the generator matrix is equal to the number of input bits plus the number of shift register bits, which is denoted by \(n\) + \(m\). The degree of each row of the generator matrix is equal to the number of feedback connections from the shift registers to that output bit [25]. In the previous example, the generator matrix has two rows, so \(k\) = 2. It has three columns, so \(n\) + \(m\) = 3. Since the first row has degree 1 and the second row has degree 2, there are two feedback connections in total, so \(m\) = 2. Therefore, \(n\) = 1. The memory required to store the generator matrix **G** of the Convolutional code is dependent on the size of the matrix. In general, the size of the matrix is determined by the rate and constraint length of the code. To provide an example, let us consider a rate 1/2 Convolutional code with s = 64 states (_m_ = 6 bits) and an LDPC code with rate and a block length of N = 2048 bits. In this case, the generator matrix would have 2 rows and 12 columns. This is because each input bit is mapped to two output bits, and the encoder uses the current input bit as well as the five previous input bits to generate each pair of output bits. Therefore, the generator matrix will be 2 \(\times\) 7. This is because \(k\) = 2 (the number of output bits per input bit) and \(n\) = 1 (the number of input bits). The number of columns is \(n\) + \(m\), which is 1 + 6 = 7. The size of memory needed for saving this matrix in the Convolutional code depends on the data type and the format of the matrix.