# Research and Implementation of High-Efficiency and Low-Complexity LDPC Coding Algorithm
## 2 LDPC Code in the IEEE802.16e Standard
### High-Efficiency and Low-Complexity Coding Algorithm (HE-LC)  
When the standard coding algorithm generates CWs via a generator matrix, the deficiency lies in the check matrix's sparsity, which is affected during the Gaussian elimination process. Consequently, the coding complexity is directly proportional to the square of the code length, consuming enormous hardware resources and imposing an excessive time delay [12; 13; 17; 18]. Nevertheless, employing the recursive-iterative coding algorithm allows for the check vector's iterative formula to be obtained through the check matrix's quasi-double-diagonal structure, which aggravates the computational complexity of the check matrix processing by simplifying the coding process. Based on the research and analysis of the check matrix, we developed a high-efficiency and low-complexity coding algorithm. Specifically, through the row-column replacement, an approximate upper triangular structure (Figure 1) appears in the middle submodule of the check matrix. Next, we partitioned the check matrix so that the upper triangular matrix of the middle submodule forms an independent submatrix. Thereby, efficient iterative coding is evident for this submatrix's special structure, guaranteeing through preprocessing the check matrix's sparsity and enabling iterative coding. In Figure 1, \(A\) is an \((m-g)\times k\) matrix, \(T\) is an \((m-g)\times(m-g)\) upper triangular matrix, \(B\) is an \((m-g)\times g\) matrix, \(C\) is a \(g\times k\) matrix, \(E\) is a \(g\times(m-g)\) matrix, and \(D\) is a \(g\times g\) matrix. The basic check matrix is given as:  
\[H_{b}= \begin{bmatrix}A_{(m_{b}-g)\times k_{b}}&T_{(m_{b}-g)\times(m_{b}-g)}& B_{(m_{b}-g)\times g}\\ C_{g\times k_{b}}&E_{g\times(m_{b}-g)}&D_{g\times g}\end{bmatrix}  \tag{19}\]  
In the basic check matrix \(H_{b}\) with a standard code rate of \(2/3\), we set \(n_{b}=24\), \(m_{b}=8\), \(k_{b}=16\), and \(g=1\). The post-transformation matrix is obtained by left multiplicative the basic check matrix \(H_{b}\) with a full-rank square matrix \(\left[ \begin{array}{cc}I_{m-g}&0\\ -ET^{-1}&I_{g}\end{array} \right]\):  
\[\hat{H_{b}}= \begin{bmatrix}A&T&B\\ -ET^{-1}A+C&0&-ET^{-1}B+D\end{bmatrix}  \tag{20}\]  
In Equation (20), \(I_{m-g}\) and \(I_{g}\) represent the unit matrices of size \((m-g)\times(m-g)\) and \(g\times g\), respectively. The CW vector is set to \(c=[u\quad p_{1}\quad p_{2}]\), where \(u\) is an information bit, and \(p_{1}\) and \(p_{2}\) are check bits. The following are obtained through the check equation \(H_{b}\cdot c^{T}=0\Leftrightarrow\hat{H_{b}}\cdot c^{T}=0\):  
\[\left\{ \begin{matrix}Au^{T}+Tp_{1}^{T}+Bp_{2}^{T}=0\\ (-ET^{-1}A+C)u^{T}+(-ET^{-1}B+D)p_{2}^{T}=0\end{matrix} \right. \tag{21}\]  
Figure 1: Approximate upper triangular structural form of a check matrix. Namely:  
\[p_{2}^{T}=-\varphi^{-1}(-ET^{-1}A+C)u^{T} \tag{22}\]  
\[p_{1}^{T}=-T^{-1}(Au^{T}+Bp_{2}^{T}) \tag{23}\]  
where \(\varphi=-ET^{-1}B+D\) is a \(g\times g\) full-rank square matrix (to be replaced by a \(z\times z\) square matrix). The structural features of the \(\varphi\) matrix are studied and analyzed, followed by MATLAB-based simulation, revealing that the \(\varphi\) matrix preserved a unit matrix within a binary field. For the LDPC code utilizing a standard code rate of 2/3, the MATLAB simulation results of the \(\varphi\) matrix are illustrated in Figure 2.  
Through comprehensive analysis, the following are obtained:  
\[p_{2}^{T}=-(-ET^{-1}A+C)u^{T} \tag{24}\]  
\[p_{1}^{T}=-T^{-1}(Au^{T}+Bp_{2}^{T}) \tag{25}\]  
Adopting the high-efficiency and low-complexity coding algorithm in the coding process, the computational complexity of \(p_{2}\) and \(p_{1}\) is \(o(n+g^{2})\) and \(o(n)\), respectively. Hence, the proposed algorithm mitigates the implementation complexity while achieving an efficient coding performance. Moreover, it reduces the coder's hardware needs, facilitating hardware implementation.