Research On The Construction Method Of QC-LDPC Codes Based On Cyclic Difference Families

Low-Density Parity-Check（LDPC）codes are a kind of classical linear block codes,whose error correction performance is extremely close to the Shannon’s capacity limit.Due to the sparsity of the parity check matrices of LDPC codes,they also have the advantages of low encoding/decoding complexity and convenience of hardware implementation.Quasi-Cyclic（QC）-LDPC codes are a class of structured LDPC codes,whose encoding can be achieved by simple linear shift registers due to the quasi-cyclic regularity in their parity check matrices.And since the required storage space is less and the complexity of the hardware implementation is lower,QC-LDPC codes have already become the research focus in channel coding field.Cyclic Difference Family（CDF）is a very important design theory among combinatorial mathematics,Cyclic Difference Set（CDS）is a subclass of CDF,and which both have been well applied in the field of channel coding.In this thesis,the construction methods of QC-LDPC codes are studied based on CDF and CDS,the main research works are presented as follows:1.To cope with the issue that the complexity of puncturing is very high due to the large size of the parity check matrices of variable-rate LDPC codes,a novel construction method of variable-rate Type-I QC-LDPC code without 4-cycles based on CDF is proposed.In this method,the base matrix is constructed by flexibly selecting the parameters t and k from CDF,thus we can acquire a class of variable rate Type-I QC-LDPC code.The base matrix P is composed of circulant matrices and identity matrices,so the storage complexity of this Type-I QC-LDPC code is only related to the first row in each circulant matrix and all“1”in each identity matrix among base matrix P,which has greatly reduced the required storage space of the matrix H.Correspondingly,it also has reduced the complexity of hardware implementation.Simulation results show that at the Bit Error Rate（BER）of 10^-5,the Net Coding Gain（NCG）of the proposed Type-I QC-LDPC codes with the code rate of 2/3 have both been improved about 0.1dB than those of randomly constructed Mackay codes and Progressive Edge Growth（PEG）codes.2.By researching on Type-II QC-LDPC code,it is found that the Type-II QC-LDPC code has a higher minimum distance upper limit than the Type-I QC-LDPC code,consequently its anti-interference ability is better.To cope with the issue that the existence of Weight-2 Circulant Matrices（W2CM）in parity check matrix of Type-II QC-LDPC code inevitably makes the Tanner graph be easier to have short cycles,which affects the convergence of iterative decoding,a novel construction method of girth-8Type-II QC-LDPC codes based on perfect CDS is proposed.The parity check matrices constructed by the proposed method consist of weight-0 zero matrices,weight-1 identity matrices and W2CM,which holds the advantage of the higher upper bound for the minimum distance and makes the error correction performance of the codes better.In addition,the Tanner graphs of these codes have no 4-cycles and 6-cycles,and thus they have the characteristics of the excellent decoding convergence in high signal-to-noise ratio region.Simulation results show that under the same conditions that the BER is 10^-5and the code rate is 0.5,the NCG of proposed girth-8 Type-II CDS-QC-LDPC（2184,1092）code has been respectively improved 0.39dB and 0.11dB than those of the girth-6Type-II CDS-QC-LDPC（2212,1108）code and the girth-8 Type-I GCD-QC-LDPC（2200,1100）code constructed by the construction method based on Greatest Common Divisor（GCD）.In addition,the NCG of proposed girth-8 Type-II CDS-QC-LDPC（6056,3028）code has been respectively improved 0.38dB and 0.12dB than those of girth-6Type-II Sidon-QC-LDPC（6056,3028）code constructed by the construction method based on Sidon sequence and the girth-8 Type-I GCD-QC-LDPC（6100,3050）code.3.To cope with the issue that the encoding complexity of Type-II QC-LDPC codes is high when using the traditional encoding algorithm based on generation matrix,a novel construction method of irregular Type-II QC-LDPC code based on perfect CDS is proposed.The parity check matrices constructed by the proposed method consist of weight-0 zero matrices,weight-1 circulant permutation matrices and W2CM.The W2CM in parity check matrices can allow a higher minimum distance to be achieved,which will make the error correction performance of codes better.The girth in Tanner graphs is at least 6,thus they have the excellent decoding convergence characteristics.In addition,since the parity check matrices have the quasi-dual diagonal structure,the fast encoding can achieve by taking directly advantage of the parity check matrices,which has reduced the encoding complexity effectively.Simulation results show that the NCG of proposed irregular Type-II CDS-QC-LDPC（1098,549）code,compared with the regular Type-II CDS-QC-LDPC（1092,546）code and the Type-I APS-QC-LDPC（1008,504）code constructed by the construction method based on Arithmetic Progression Sequence（APS）,has been respectively improved 0.39dB and 0.22dB under the same conditions that the BER is 10^-5and the code rate is 0.5.And under the same conditions that the BER is 10^-5and the code rate is 0.67,the NCG of irregular Type-II QC-LDPC（4977,3318）code has respectively been improved 0.59dB and 0.31dB than those of the QC-LDPC（4665,3114）code constructed by removing row-blocks of parity check matrix and the regular Type-II CDS-QC-LDPC（5226,3486）code.