# Blind Recognition of Forward Error Correction Codes Based on a Depth Distribution Algorithm
## 6 Depth Distribution Algorithm Based on Parallel Convolutional Turbo Codes
### Recognition Algorithm of Parallel Convolutional Turbo Coding
At the same time, similar to the selection of \(M(n_{e},v_{e},D_{e},l_{e})\), we select the row vector from \(C^{(k+i)}(n_{e},v_{e},D_{e},l_{e})\) whose row sequence number is \((vD+1)\) to constrain matrix \(C^{*(k+i)}(n_{e},v_{e},D_{e},l_{e})\). For a more general case, when the interleave length is \(k(D+1+a)\), the coded reconstruction equation can be obtained according to Equation (39):  
\[\left[ \begin{array}{ccccc}m^{l-1}&\cdots&m^{0}&c_{k+i}^{D+a}&\cdots&c_{k+i}^ {0}\\ m^{2l-1}&\cdots&m^{l}&c_{k+i}^{2(D+a)+1}&\cdots&c_{k+i}^{D+a+1}\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots\end{array} \right]X=0 \tag{40}\]  
In addition, for some encoders, the interleave length is not equal to \(k(D+1+a)\) and the unity between the analysis and the identification process can be realized by solving the least common multiple \(l^{*}\) between them. So far, the reconstruction of Turbo code is solved skillfully by using a linear block code reconstruction algorithm, and the recognition of the Turbo code can refer to the basic algorithm of the linear block code. In order to realize the code type classification, the correlation between the depth distribution characteristics of the Turbo code word matrix and the linear block code word matrix is also analyzed. According to previous studies on linear block codes and convolutional codes, the depth distribution characteristics of the code word matrix of Turbo codes are related to a parameter estimation of the code length, code rate, starting position and interleave length. Under different parameter estimation conditions, the depth distribution is significantly different. The parameter estimation of Turbo is also realized by reconstructing the equivalent linear block code, so the code length \((n_{1}+n_{2}-k)\) and rate \(k/(n_{1}+n_{2}-k)\) can be estimated by the recognition method of linear block code length and the code rate. Therefore, the effect of interleave length estimation on the depth distribution characteristics of the Turbo code word matrix is an important research object. We assume that the codeword matrix stores \(\theta(n_{1}+n_{2}-k)l^{*}\) bits of data altogether, and \(\mu\) is a positive integer. For parameter estimation of RSC1, RSC2 and interleave in encoder, when \(l^{*}=k(D_{1}+1+a_{1})=k(D_{2}+1+a_{2})\), if \(n_{e}=\theta(n_{1}+n_{2}-k)l^{*}\), the linear independent vectors in RSC1 and RSC2 are completely preserved. Under this condition, the number of information bits in the sequence output by RSC1 is \(\theta k(D_{1}+1+a_{1})\) and the ones by RSC2 is \(\theta k(D_{2}+1+a_{2})\). According to Equations (33) and (34), the total number of check bits in the code word matrix is \(\theta((n_{1}-k)(D_{1}+1+a_{1})+(n_{2}-k)(D_{2}+1+a_{2}))\). If \(n_{e}=(n_{1}+n_{2}-k)(D_{1}+1+\varphi)\), \(\varphi\) is a positive integer and \(n_{e}\
eq\theta(n_{1}+n_{2}-k)l^{*}\). Under this condition, based on the asymmetry of the matrix, all linear independent vectors in RSC1 are completely retained, while some linear independent vectors in RSC2 are not retained, which makes the depth distribution of the codeword matrix change.