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Journal of the China Railway Society

The Essential Conditions of Quasi-cyclic LDPC Codes without Girth 4

Yang Xiao

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

The essential conditions for quasi-cyclic low-density parity-check codes(QC-LDPC codes) to have no Girth 4 are given.This solves the problem of designing without-girth-4 QC-LDPC codes of any length.Adjusting the dimension and shift factors of the circulant sub-matrices of the given sparse parity-check matrices according to the proposed theorems,the QC-LDPC codes without girth 4 are constructed.The matrice equation set for girth 4 check is established.Through setting the dimension and shift factors of the circulant sub-matrices,we can check whether the QC-LDPC codes have Girth 4 or not.We can multiply the dimension of the circulant sub-matrices to get the QC-LDPC codes without Girth 4 of different lengths.Our theorems keep the QC-LDPC codes without Girth 4 along with increasing of the code length.Compared to the previous algorithms,the proposed test algorithm decreases computation,shortens the test time and reduces the complexity of structuring the parity-check matrix.Experimental results indicate that the designed QC-LDPC codes achieve good bit error rate(BER) performance.

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The essential conditions for quasi-cyclic low-density parity-check codes(QC-LDPC codes) to have no Girth 4 are given.This solves the problem of designing without-girth-4 QC-LDPC codes of any length.Adjusting the dimension and shift factors of the circulant sub-matrices of the given sparse parity-check matrices according to the proposed theorems,the QC-LDPC codes without girth 4 are constructed.The matrice equation set for girth 4 check is established.Through setting the dimension and shift factors of the circulant sub-matrices,we can check whether the QC-LDPC codes have Girth 4 or not.We can multiply the dimension of the circulant sub-matrices to get the QC-LDPC codes without Girth 4 of different lengths.Our theorems keep the QC-LDPC codes without Girth 4 along with increasing of the code length.Compared to the previous algorithms,the proposed test algorithm decreases computation,shortens the test time and reduces the complexity of structuring the parity-check matrix.Experimental results indicate that the designed QC-LDPC codes achieve good bit error rate(BER) performance.
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.
To solve the problem of the girth 4 which will lead to a serious decrease in the performance of quasi-cyclic low-density parity-check(QC LDPC) codes,this paper presents the necessary and sufficient conditions of Tanner codes without girth 4 in the form of several theorems.These theorems can be used as the constraint conditions for the construction of Tanner codes.To construct the Tanner codes without girth 4 needs to adjust the dimension and the shift factors of the circulant matrices of the given sparse parity-check matrices according to the proposed theorems.Examples are provided to verify the theorems,and computer simulation results show that the Tanner codes obtained by proposed constructions achieve good BER performance compared with that of randomly constructed LDPC codes.
This letter presents a necessary and sufficient condition for a class of quasi-cyclic low-density parity-check (QC LDPC) codes without girth four. Girth-four property of a class of QC LDPC codes is investigated. Good QC LDPC codes without girth four can be constructed by selecting proper shifting factors according to the proposed theorems. Examples are provided to verify the theorems. The simulation results show that the QC LDPC codes without girth four achieve a better BER performance compared with that of randomly constructed LDPC codes.
A design algorithm of regular check matrix for Low-Density Parity-Check(LDPC)codes is proposed in this paper.The proposed approach is to design 3 sub-matrices,and each of the sub-matrices is combined by the cyclic shift matrices of the identity matrix,then to combine these sub-matrices into a desired parity check matrix according to our approach.Using the check algorithms of the short cycles to find the combinations of shifting operators by which the parity check matrix desires has no girth- four and no girth-six.The simulation results in AWGN channel show that the BER performance of these codes obtained by the generator matrix derived from the proposed check matrices to encode the random information bits is as good as that of random LDPC codes.
This paper presents a novel regular Quasi-Cyclic (QC) Low Density Parity Check (LDPC) codes with column-weight three and girth at least eight. These are designed on the basis of combinatorial design in which subsets applied for the construction of circulant matrices are determined by a particular subset. Considering the non-existence of cycles four and six in the structure of the parity check matrix, a bound for their minimum weight is proposed. The simulations conducted confirm that without applying a masking technique, the newly implemented codes have a performance similar to or better than other well-known codes. This is evident in the waterfall region, while their error floor at very low Bit Error Rate (BER) is expected.
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.