Construction Of Low-Density Parity-Check Codes Based On Density Evolution And Finite-Length Analysis

As a kind of channel coding technique with wide application, LDPC codes have been adopted in DVB-S2 and 802.16e. Since the performance can achieve or exceed turbo codes while encoding and decoding complexity is quite low due to the sparseness of parity matrix, LDPC codes have become the focus of coding theory. In comparison with regular LDPC codes, irregular codes can achieve better performance in terms of noise threshold, which is largely affected by degree distribution pair. Using density evolution, the performance of codes with same degree distribution can be determined. Thus, degree distribution pair can be optimized on the basis of density evolution, which makes it possible to find good codes. However, optimal degree distributions always result in poor error floor, so it is necessary to add constraint on degree distribution pairs and take finite-length effects into consideration. On the other hand, for particular degree distribution pair, though codes under random construction may have similar performance, the parity matrix should have well-designed structure for the sake of implementation. With study of above issues, LDPC codes with good performance and low implementation complexity can be constructed and applied in communication systems for error-correcting.In this paper, firstly decoding algorithms including belief propagation, min-sum algorithm is analyzed under density evolution. Then density evolution of layered decoding is deduced, and performance comparison of various decoding algorithms is made in terms of threshold and convergence. Thus optimal scale factor for modified min-sum algorithm is obtained and it is shown that with layered decoding, iteration times can be reduced by almost half.In addition, with finite-length analysis, the impact on error floor by structure of parity matrix is studied. Since error floor is mainly caused by cycles with low extrinsic message degree, some constraints on variable degree distribution are put forward, which can be used to optimize degree distribution pair.Finally, with consideration of encoding and decoding complexity, a construction method of irregular LDPC codes is presented. With constraint on number of degree-2 variable nodes and approximate cycle extrinsic message degree detection, cycles that tend to be stopping sets are avoided. As a result, the constructed irregular codes remain to have good threshold performance and the problem of high error floor is properly solved.With the above studies, it can be concluded that density evolution and finite-length analysis are two key theories for analysis and construction of LDPC codes. From the point of threshold and error floor, good codes can be obtained by properly select degree distribution pair and parity matrix structure.