Research On Non-binary Low-Density Parity-Check Codes

Low Density Parity Check (LDPC) codes can achieve excellent error-correcting performance close to the benchmark predicted by the Shannon theory, and it attracts much more attention in recent years. While the nonbinary LDPC code is a new branch of the research of LDPC codes, many researchers focus on the code construction, the performance analysis and the optimization algorithm, which have broad prospects and profound academic significance during the research of nonbinary LDPC codes.In communication systems in future, as different multimedia services (e.g., voice, video conference, streaming media, web browsing) have different quality of service (QoS) requirements and the wireless channels are constantly timevarying, the physical and link layer parameters of an adaptive communication system such as the length of a data frame, the code rate, and the block length need to be adjusted accordingly based on traffic characteristics and channel conditions. Meanwhile, existing literature mostly focused on the computation of the minimum distance of binary codes, however, the methods on computing the minimum distance of nonbinary LDPC codes haven’t been proposed yet. Moreover, most design methods on LDPC codes have been the construction of the parity check matrix (H matrix) with large girth, but there are already many good nonbinary LDPC codes with small girth. From this viewpoint, the girth could not be considered as a good parameter to be optimized for LDPC codes. Therefore, there is one active research direction on the construction of LDPC codes aiming to seek a more rational optimization way than maximizing the girth.Quasi-cyclic LDPC (QC-LDPC) codes are particularly appealing to practical systems as the quasi-cyclic structure of the H matrix allows for linear time encoding using only shift registers, renders efficient routing for decoding implementation, and enables the storage of the coding matrix with only a few memory units. So this work constructs several nonbinary LDPC codes with high performance. Firstly, the dissertation proposes the design method to construct nonbinary QC-LDPC codes with continuously variable length, and obtains multiple rates codes by the shortening/appending method. Secondly, the construction method for multiple rates nonbinary QC-LDPC codes is elaborated by splitting the H matrix. Thirdly, the survey proposes to use the nonbinary Galois field as one degree of freedom to design codes of different rates by fixing either the information or the block bit length, tailored to different scenarios in wireless applications.It is difficult to find out the minimum distance of LDPC codes. In this dissertation, the research aims to compute the minimum distance of nonbinary LDPC codes to tackle this problem. Based on the characteristics of nonbinary LDPC codes, this work modifies the binary Nearest Nonzero Codeword Search (NNCS) method, including the modified error impulse pattern, the dithering method, and the nonbinary decoder. Numerical results on the estimated minimum distances show that a code’s minimum distance can be increased by careful selection of nonzero elements of the H matrix, or by increasing the mean column weight, or by increasing the size of the Galois field. These results support the observations that have been made based on simulated performance in the literature.During the research, it has been recognized that the performance of QC-LDPC codes does not depend much on the girth. So the dissertation uses an estimated minimum distance upper bound as the design constraint instead of the girth to construct the H matrix of nonbinary QC-LDPC codes. Since large minimum distances often translate to lower error floors, the proposed method provides an alternative appealing design methodology for nonbinary QC-LDPC codes.