Circulant Search Algorithm And Optimization Of The Construction For QC-LDPC Codes

Low-density parity-check codes, discovered by Gallager in1962, were ignoredbecause of high encoding complexity until1990’s. After the discovery of turbo, LDPCcodes showed the excellent performance close to the Shannon limit as turbo. LDPC codeswith Quasi-Cyclic structure are important branch of LDPC codes. Due to the specialstructure of check matrix, QC-LDPC codes can be encoded using simple feedback shiftregisters with linear complexity. The research point of the paper is about constructingregular/irregular QC-LDPC codes that achieve good performance.First, a circulant search algorithm(CSA) for the construction of QC-LDPC codes ispresented in this paper. Given a mother matrix, the algorithm can search proper cyclic shiftparameters to construct a QC-LDPC code with large girth. For a (2, k) LDPC code, the girthdetermines min-distance derectly. To get LDPC codes with larger girth, the paper gives arecursive design based on (k, k) LDPC codes with girth g to construct (2, k) LDPC codeswith girth2g. By the proposed method, we constructed a (2,3) LDPC code with girth g=36.Given a irregular mother matrix, CSA can derectly construct a irregular QC-LDPC codewith desired girth, which has a large number of degree-2bit nodes. It’s possible to havelow-weight codewords consisting of degree-2bit nodes and a small number of degree-3bitnodes. After construction analysis of low-weight codewords, a optimization algorithm ofCSA is presented to search irregular QC-LDPC codes with less low-weight codewords,which can increase the min-distance to lower the error floor and improve performance.The simulation results on AWGN channel show that the QC-LDPC codes circulantsearch algorithm constructs have excellent performance. And the comparison betweenoptimized irregular codes and unoptimized irregular ones shows that the improvement ofoptimization is obvious.