Construction Of LDPC Codes Based On Rectangular Lattices

Low-density parity-check(LDPC) codes are an important group of linear block codes. Since the LDPC codes were demonstrated to perform very close to the Shannon limit, they become a hot topic in coding theory. The codes have good error correcting performance. Its decoding algorithm, known as the sum-product or belief propagation algorithm, has an inherent parallel characteristic which make them easy for high-speed parallel processing. At the same time, LDPC codes have high error correction ability even under condition of high rate, which leads a wide range of applications in future communication system and storage system. Quasi-Cyclic(QC) Low-Density Parity-Check(LDPC) codes represent an important class of codes within the family of LDPC codes. The check matrix of these codes consist of small zero squares and small cyclic matrix, the special structure leads the lower encoding and decoding complexity.In the construction of QC-LDPC codes, Vasic et al. have constructed matrix without 4-cycle and 6-cycle construction, thus it can construct codes with girths at least eight. We studied the gragh structure on rectangle lattices and find out the limitations of this algorithm. Firstly, the algorithm is available if and only if the column weight is three, namely, it can not construct codes with higher column weight. And secondly the constructed code have girths no more than eight. In this article, we analysed the graph structure on rectangular lattices and find out some constrains between certain trapping sets and the slopes of lines constructed them. Some small trapping sets would be removed out once meet these constrains. In addition, we got a set of constrains about polygons in analysis of cycles and slope of lines on rectangular lattices. As long as we find out slopes dissatisfy with these constrains, we can construct a series of LDPC codes with special parameters. In this article, we constructed three kinds of LDPC codes with large girth. The first kind of codes has column weight larger than three with girth stay for eight. The second kind of codes is a kind of(3, k) code whose column weight is three with girth ten or twelve. And the last kinds is a kind of(3, k) code with girth fourteen.Finally, we compared the performance between the codes constructed by our improved algorithm and the original algorithm. It can be seen from the simulation results that, since the column-weight or the girth has been enlarged, some small trapping sets has been eliminated consquently, the code shows a lower error floor, that is, the performance is better.