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Construction of LDPC Code based on Permutation Matrices Optimization

Dong Zheng

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

The presence of circles would lay a great impact on the quasi-cyclic LDPC decoding.Therefore,the parity check matrix with good girth distribution is beneficial to the performance of code.This paper first makes a analysis on girth distribution of the check matrix.Then a reasonable constraint is set between the mother matrix and shift matrix,thus,by this way,the check matrix could maximize the minimum cycle length of the check matrix and the average length of the cycle.Finally,a method to construct QC-LDPC code with high girth is given.Simulation results show that this method for minimizing the short code cycle could construct good practical codes.

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Pacific-Asia Conference on Circuits, Communications and Systems•Jul 1, 2011•4 citations
The presence of circles would lay a great impact on the quasi-cyclic LDPC decoding.Therefore,the parity check matrix with good girth distribution is beneficial to the performance of code.This paper first makes a analysis on girth distribution of the check matrix.Then a reasonable constraint is set between the mother matrix and shift matrix,thus,by this way,the check matrix could maximize the minimum cycle length of the check matrix and the average length of the cycle.Finally,a method to construct QC-LDPC code with high girth is given.Simulation results show that this method for minimizing the short code cycle could construct good practical codes.
A compressing method for the construction of quasi-cyclic LDPC(QC-LDPC)codes was proposed,which used a QC-LDPC code with large girth as mother code.By compressing circulant permutation matrices of the mother code,shorter QC-LDPC codes would be generated.After optimization,the generated codes would have large girth and few cycles,whose lengths varied continuously in a large range.Analysis and simulation results show that these LDPC codes have good performance and can greatly reduce the size of code storage.
By analyzing the shapes of 4-cycles and 6-cycles of TG in parity check matrix,this paper presents a method of counting the number of 4-cycles and 6-cycles.Considering the cycles,especially the short cycles,degrade the performance of LDPC decoders,this paper then proposes a design of regular check matrix for Low-Density Parity-Check(LDPC) codes with girth 8.The proposed construction algorithm for LDPC codes is to design 3 submatrices with different given shifting functions,then combine them into one matrix,and finally expand the transposed matrix of the combination matrix into a desired parity check matrix using the identity matrix and the cyclic shift matrices of the identity matrix randomly.The simulation results in the AWGN channel show that the BER performance of these codes obtained by the generation matrix derived from the proposed check matrices in encoding the random information bits is as good as that of the random LDPC codes.
The existing construction of Quasi-Cyclic low-density parity-check (short for QC-LDPC) codes has not considered the problems of small stopping sets and small girth and small minimum code weight together, while their existences will lead to the BER performance of QC-LDPC codes to be much poorer than that of randomly constructed LDPC codes even decoding failure. To solve the problem, some theorems of the QC-LDPC codes without small stopping sets and small girth were proposed first. Then good QC-LDPC codes without small stopping sets and small girth were designed based on the row vectors of generator matrix. The small minimum code weight also can be prevented due to the new algorithm based on the row vectors of generator matrix, which is the new way to choose the shifted factors to construct the good QC-LDPC codes for the given prime. The effectiveness and the practicability of the algorithm are demonstrated by the simulation results. It is significant for us to analyze and optimize the design of LDPC codes.
The existing design of quasi-cyclic(QC) low-density parity-check(LDPC) codes has not considered the problems of short length girths and the rows' dependency.The first problem leads to the BER performance of QC LDPC codes to be much poorer than that of randomly constructed LDPC codes,while the second problem leads to the difficulty of construction of generator matrices from parity-check matrices.To solve the first problem,the constraint conditions for designing the QC LDPC codes are proposed,and the dimension and shift factors of the circulant matrices of the given sparse parity-check matrix are adjusted according to the test results of both girth 4 and girth 6.To solve the second problem,an approach of irregular QC LDPC codes is proposed.The proposed approach is to replace some sub-matrices by zero matrices and identity matrices at special positions in the given sparse parity-check matrix so as to get a nonsingular square matrix for the construction of the generator matrix.Though adopting bidiagonal submatrices in parity-check matrix can solve the rows' dependency problem,many code words with low weights will occur,which leads to the BER performance can not be better by increasing code lengths.Examples are provided for the proposed design of QC LDPC codes,and computer simulation results show that the proposed QC LDPC codes achieve good BER performance.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.
Due to quasi-cyclic structure of parity-check matrices, Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes reduce hardware requirements, and thus they have a higher practical value than random LDPC codes. This paper proposes an improved construction method of QC-LDPC based on PEG Algorithm, and the girth of the QC-LDPC codes has an increase of 2. Firstly, a base matrix is generated by the PEG algorithm, and then the base matrix is expanded using the unit circle permutation matrix, at the same time, the short loop of the base matrix is eliminated. This approach has the advantages of the PEG algorithm and the QC-LDPC codes, and simulation results show that bit error rate (BER) performance of the QC-LDPC codes presented in this paper is the same as or exceeds the one of randomly constructed LDPC codes.