Study On The Algebraic Constructions And Decoding Algorithms Of LDPC Codes

Being a powerful class of error-correcting codes based on graphical models and iterative decoding algorithms, low-density parity-check (LDPC) codes have recently received much attention due to their capacity approaching performance and relatively low decoding complexity.This dissertation is intended to investigate the algebraic construction and the iterative decoding algorithm of LDPC codes. The main results are summarized as follows.1. A method of constructing quasi-cyclic (QC) LDPC codes of large length by combining QC-LDPC codes of small length as their component codes via the Chinese Remainder Theorem (CRT) is researched. Due to the similar structural property of the component codes, there exist a lot of short cycles in the Tanner graph of the designed code and the performance is not good enough with iterative decoding. A generalized CRT method is proposed to improve the CRT combining method and design much more and better QC-LDPC codes when the parity check matrices of the component codes are given. By permuting the block rows of the parity check matrices of component codes, a lot of short cycles in the Tanner graph of the designed code can be eliminated and a large number of QC-LDPC codes with better performance can be designed. By loosening the condition for determining the intermediate parameters, a much larger class of QC-LDPC codes with better performance can be designed.2. Based on the structural properties of Euclidean geometry, an algebraic method is proposed to construct a class of QC-LDPC codes with circulant permutation matrices. This class of QC-LDPC codes contains smaller number of short cycles in the Tanner graph and has similar performance with the existing QC Euclidean geometry LDPC codes.3. The distributions of codewords with minimum weight of the existing QC Euclidean geometry LDPC codes are analyzed and a sufficient condition for a codeword to have minimum weight is found. A construction method, which can reduce the number of minimum weight codewords satisfying this sufficient condition, is proposed. The new QC-LDPC codes have much lower error floor and perform better at low BER.4. Some weighted bit-flipping decoding algorithms based on soft information are studied and a modified bit-flipping decoding algorithm with significantly low complexity is proposed. The new algorithm has significantly high decoding speed and short decoding delay. 5. Based on the ability of continuous transmission for LDPC convolutional codes, an iterative feedback belief propagation decoding algorithm is proposed. The new algorithm utilizes the most current variable information of the past code bits to activate the variable nodes more efficiently by applying feedback information at each decoding iteration. The proposed algorithm with small number of iterations can achieve better performance than that of the existing algorithm, while the decoding complexity and latency are effectively reduced.6. Inspired by conclusion 5, a fast convergence decoding algorithm for LDPC convolutional codes is developed. A better performance is achieved, while the complexity and decoding delay are effectively reduced, which makes LDPC convolutional codes be more suitable for practical applications.