Research On The Minimum Distance And Application Of Low-Density Parity-Check Codes

With the increasing demand for high-speed services in mobile communication systems, people are putting more emphasis on forward error correction (FEC) codes. LDPC codes are a kind of FEC codes which have the near-Shannon-limit performance and low decoding complexity. So it has become a hot spot in the field of channel coding. In this dissertation, some issues of LDPC codes are investigated, including the basic theory, the minimum distance and its application. The main content involves the representations of LDPC codes, the construction of sparse parity-check matrix, LDPC decoding methods, measurement of the minimum distance and the code weight distribution and applications of LDPC codes in MIMO-OFDM systems. The main work is as follows:1. The development of channel coding is summarized from theory to practice, and the fundamentals of linear block codes (LBC) are investigated, including the weight of LBC, the distance and the minimum distance. On that basis, several common minimum distance bounds of LBC are given.2. The construction and decoding algorithms of LDPC codes are discussed. Based on the introduction of the representations of LDPC codes (sparse parity-check matrix and Tanner graph) and the degree distributions, PEG algorithm is considered to construct LDPC codes. Furthermore, the construction methods in IEEE802.11n and WiMAX802.16e standard are also described. As for decoding, the LLR BP algorithm and Min-sum algorithm of LDPC codes are expressed in this dissertation.3. The minimum distance and the code weight distribution are important indicators for analyzing the performance of LDPC codes in theory, while the approximately nearest codeword (ANC) method is a commonly used algorithm to calculate the minimum distance of LDPC codes. In this paper, we describe in detail the implementation of the ANC method, especially how to search the codeword which is approaching the minimum weight codeword. Based on this, the minimum distances and low code weight distributions of several LDPC codes in WiMAX802.16e standard are given. Furthermore, the performance of LDPC codes at high signal-to-noise ratio is analyzed using union bound, and the upper bounds of LDPC codes are given. The results verify the excellent error performance of LDPC codes at high signal-to-noise ratio. It meets the demands of low-error rate needed in WiMAX physical layer.4. A MIMO-OFDM system using LDPC encoding and decoding method is designed, and the performance of some groups of LDPC codes in 802.16e standard is analyzed. It reconciles the results of their minimum distances.