Theory Research And Design Of Good LDPC Codes

Low-Density Parity-Check (LDPC) Codes are a class of channel codes based on graphs and iterative decoding whose performance is very close to the Shannon limit with low complexity and have strong error strength. Because of their outstanding performance, good encoding algorithm which has almost linear time complexity, decoding algorithm which could be implemented in parallel architecture, and the wildly application future, they have become one of the hottest topics in coding field today. In this paper, the theory and design of LDPC codes are studied, which involves algebra principles, block codes principles, encoding/decoding, and construction of sparse parity-check matrix,. we mainly do the research on the encoding optimization of LDPC codes.First of all, based on the minimum distance and girth of LDPC codes, this paper analyzes the influence on the performance of codes brought form the existence of short cycle such as 4-cycle and 6-cycle,and deeply studys the shapes of the cycles of TG in parity check matrix, then presents a kind of algorithms with low computational complexity for counting the number of short cycles .Then, by combining this algorithm with the presented design of quasi-cyclic (QC) LDPC codes, this paper propose the constraint conditions for the construction of the QC LDPC codes, it needs adjust the dimension and the shift factors of the circulant matrices of the given sparse parity-check matrices according to the test results of short girths. By this way, we can obtain the QC LDPC codes without 4-cycle,sometimes even without 6-cycle ,and the performance of the QC LDPC codes is quite good.Finally, this paper proposes two approaches of irregular QC LDPC codes. The first approach is to replace some sub-matrices by zero matrices and identity matrices at special positions in the given sparse parity-check matrices, to get a nonsingular square matrix for the construction of generator matrix. The second approach is permute the columns of the designed parity-check matrix to obtain a sub-matrix with all 1s at its diagonal line, then delete some 1 elements below the diagonal line to get a non-singular square matrix for the construction of generator matrix. Examples are provided for the proposed design of QC LDPC codes, and computer simulation results show that the QC LDPC codes by proposed constructions achieve good BER performance.