Research On Enumeration Of Cycle Structures And Construction Of Matrix With Girth-8/10/12 Of QQC-LDPC Codes

Low-density parity check(LDPC)codes is the best error correcting code which approaches Shannon's capacity limit.LDPC code has good asymptotic performance,with decoding complexity linearly proportional to the codes length.The girth of Tanner graph(shortest cycle length)corresponding to parity check matrix is one of the most significant factors that influence the performance of LDPC code,the major contents of this paper is researching the structures of 6/8/10-Cycles and construction of shift value matrix with girth-8/10/12 of Quasi-Cyclic LDPC code.The major contents are as follows:First,this paper studied how to recognize the existed short cycles in the shift value matrix of QC-LDPC code.On the basis of the necessary and sufficient conditions for the existence of 21 cycle in the shift value matrix,a short cycle recognition algorithm based on closed path is proposed,the complexity of recognition 21 cycle in a m×n shift value matrix is O(ml+1nl+l).In order to find all the structures of short cycle accurately,avoid duplicate and invalid judgment,a short cycle structures enumeration method based on sub-matrix partition and difference matrix is proposed.In addition,a method of fast enumerate the discriminant expressions is proposed from the combination of sub-matrix partition and closed path traversal.Then,constructing QC-LDPC codes with high rate and large girth on the basis of enumerating the 6/8/10 cycle structures,a traversal search algorithm is proposed for finding the minimum p with given girth.Under conditions of the first row and column of the shift value matrix are all zero element,the number of attempts in the worst case of the traversal search algorithm is Ap-1(m-1)*(n-1)Based on the study of the cycle structures,the second row of the shift value matrix to be searched can be constructed by the difference set in the combinatorial mathematics withγ=1.A simplified search algorithm based on the difference set with γ=1 is proposed,and the number of attempts in the worst case is reduced to Ap-n(m-2)*(n-1).