Construction Of LDPC Codes Based On Circulant Permutation Matrices

Low-density parity-check (LDPC) codes are a class of Shannon limit approaching codes. Their decoding complexities linearly grow with their lengths. LDPC codes are defined by their parity-check matrices. The structures of their parity-check matrices directly affect their performances. According to the construction methods, LDPC codes can be roughly divided into two types:random and structured. The latter has become a research hotspot in recent years. In this thesis, two new design approaches for constructing structured LDPC codes based on circulant permutation matrices (CPMs) have been proposed.First, a matrix can be constructed based on two intersecting lines and their intersecting point of Euclidean geometry (EG). Then a class of regular quasi-cyclic (QC) LDPC codes can be obtained by replacing each field element in the matrix with a relevant CPM. Compared with girth-6 EG-LDPC codes, these codes have a girth of at least 8, which is of great benefit to the improvement of their performances. Furthermore, these codes possess the quasi-cyclic structures and hence their encoding complexities can be reduced.Secondly, an array can be obtained by properly arranging CPMs of different sizes. Then a submatrix can be extracted from the array such that its column weights are no less than 3. This submatrix is taken as the parity-check matrix. This contruction method is very simple, and is easy to satisfy the row-column constraint of the parity-check matrix. A class of irregular LDPC codes with different row weights or various row and column weights can be obtained through this method. Since the sizes of CPMs can be flexibly chosen, these codes have a wide range of rates and lengths. Since the proportion of 1 in the CPM is very small, some parity-check matrices constructed based on this method can be very sparse.The lengths of the LDPC codes constructed based on these two new methods can be from tens to thousands, and the code rates can be from 0.2 to 0.9. Simulation results show that these codes have similar or even better performances, similar fast decoding convergence speeds, and lower error floors than random Gallager and Mackay codes.