Construction Of A Class Of QC-LDPC Codes

Since the low-density parity check(LDPC)code provides near-capacity performance on many channels such as binary erasure channel(BEC),binary symmetric channel(BSC),and additive white Gaussian noise channel(AWGN),it has been more and more widely used in the fields of optical communication,mobile communication,satellite communication and information storage.An important parameter affecting the performance and determining efficiency of iterative decoding algorithms for LDPC codes,is the girth which determines the number of independence iterations.To a certain extent,the longer the girth,the better the performance.Though computer search,Fossorier constructed BC-LDPC codes with girth 8,10,12 respectively.M.Esmaeili and M.Gholami introduced a method constructing QC-LDPC codes with maximum girth 18 applying the concept of BSG and slope matrices-Later,they presented the concept of(v,b,3,1)-packing design and gave a class of QC-LDPC codes with girth 20.In this paper,we propose some new(v,b,3,1)-packing designs whose incidence matrices can be considered as mother matrices of column-weight three quasi cyclic low-density parity-check codes.The mother matrix has the following two features.First,each column has only three nonzero elements.Second,in any of the two columns,there is only one location in which there is a nonzero element.In the mother matrix,we replace the nonzero element in each column from top to bottom in the order of placement by I,Isj,Is’j.So that,we can get the parity-check matrix of QC-LDPC cede,which also can be called expansion matrix.It can be proved that the maximum length of the QC-LDPC codes constructed by this method can reach 20.The shift chain S is the vector of all shift indices.It is clear that if we confirm the shift chain S,we can get the parity-check matrix of QC-LDPC codes.On the basis of M.Esmaeili’s search algorithm,we improve the algorithm of searching shift chain S,from the original can only determine a single shift index value to determine the value of each of the two shift indices.Although M.Esmaeili gave a search algorithm for the shift chain S,but they didn’t point out how to confirm the conditions of the shift chain S.Therefore,this paper gives a recursive search algorithm to find all short cycles which lengths are less than 2g in the mother matrix,combined with the Fossorier given cycle existence condition,we obtain the discriminant of the shift index value when the length is greater than or equal to 2g in the expansion matrix,which furthers accelerate the search algorithm to find shift chain S.