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Journal of Central South University(Science and Technology)

Construction of QC-LDPC codes based on Yang Hui triangle

You ZHANG

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

For the issue of determining cyclic shift coefficients of the quasi-cyclic sub-matrix in the quasi-cyclic low-density parity-check, a method was presented based on Yang Hui triangle to compute the cyclic shift coefficients. By this method, cyclic shift coefficients could be expressed in simple analytic expressions, and cycles of length four in parity matrix were eliminated. The parity matrix is quasi-cyclic to save required memory and is prone to coding and decoding, and has high flexibility with respect to the design of code length and rate. Over an additive white Gauss noise channel and under the BP decoding algorithm, simulations show that the SNR of the QC-LDPC codes with the proposed algorithm is better than random codes close to 0.3 dB at the BER performance of 10-4 and better than the LDPC codes in DVB-S2 0.2 dB at the BER performance of 10-6. Moreover, the BER performance of the new codes is no less than the LDPC codes in IEEE 802.16e under the same conditions. Furthermore, the simulation result also indicates that by making a reasonable choice of the size of the cyclic shift matrix, the BER performance of the codeword can be improved.

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Systems engineering and electronics•Jan 1, 2009•1 citation
For the issue of determining cyclic shift coefficients of the quasi-cyclic sub-matrix in the quasi-cyclic low-density parity-check, a method was presented based on Yang Hui triangle to compute the cyclic shift coefficients. By this method, cyclic shift coefficients could be expressed in simple analytic expressions, and cycles of length four in parity matrix were eliminated. The parity matrix is quasi-cyclic to save required memory and is prone to coding and decoding, and has high flexibility with respect to the design of code length and rate. Over an additive white Gauss noise channel and under the BP decoding algorithm, simulations show that the SNR of the QC-LDPC codes with the proposed algorithm is better than random codes close to 0.3 dB at the BER performance of 10-4 and better than the LDPC codes in DVB-S2 0.2 dB at the BER performance of 10-6. Moreover, the BER performance of the new codes is no less than the LDPC codes in IEEE 802.16e under the same conditions. Furthermore, the simulation result also indicates that by making a reasonable choice of the size of the cyclic shift matrix, the BER performance of the codeword can be improved.
In this paper, we proposed a novel method for constructing quasi-cyclic low-density parity-check (QC LDPC) codes based on cyclic subgroup over GF(q) . LDPC codes constructed by this method are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN) with sum product algorithm (SPA). The main advantage is that QC LDPC codes with a variety of block lengths and rates can be easily constructed with no cycles of length four or less. Simulation results show that the proposed QC LDPC codes perform slight better than the random regular LDPC codes for short to moderate block lengths. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.
Quasi Cyclic Low Density Parity Check(QC-LDPC) code is an important branch of LDPC code research.It is a kind of constructed LDPC code with check matrix H constructed by quasi cyclic method.IEEE 802.16e proposes a parity check matrix with quasi cyclic character which makes the encoding and hardware easy to be implemented.This article will mainly introduce the encoding process and analyze the decoding performance of various decoding algorithms.Simulations show that Min-Sum(MS) algorithm compares to the Log-BP performance has approximately 0.5dB loss for high-speed rate coding.And the performance of Normalized Min-Sum(NMS) and Row Weight Matched Min-Sum(Rwm-MS) with relatively low complexity is nearly the same as that of Log-BP algorithm.
Combining the perfect cyclic difference sets with the quasi-cyclic codes,a new algorithm for constructing QC-LDPC codes based on cyclic difference sets is proposed.First,the author obtains a small initial matrix that meet the requirements of the weights based on cyclic difference sets,and then the zeros elements,non-zero elements of the initial matrix are replaced by all-zero matrices and cyclic shift matrices.For the purpose of determining the shift coefficient,the cycle length of six is eliminated by computer searching.Simulations show that,under the AWGN channel and the BP decoding algorithm,our QC-LDPC codes are better than the random LDPC codes in the bit error rate and they are more proper for hardware implementation.
Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes have practical significance in the field of engineering, thus their construction algorithm is a hot issue in the field of LDPC codes. Based on existing construction methods, particularly the method based on Progressive Edge-Growth (PEG) algorithm, a new construction method of shift matrix is proposed. It can effectively avoid the copy of cycle structures, especially short cycles in base matrix after extension. Besides, it also relieves the time cost caused by random search, and greatly improves the length of cycles when compared with algorithms based on methods such as quadratic congruence and arithmetic sequence. Those algorithms can only eliminate short cycles with length 4, while our method can provide QC-LDPC codes without length 6 at least. Cycle analysis and simulation results show that the newly constructed QC-LDPC codes have good cycle structure and error correction performance.
Quasi Cyclic Low Density Parity Check codes (QC-LDPC) is proposed to reduce the complexity of the Low Density Parity Check codes (LDPC) further while obtaining the similar performance. Our proposed QC-LDPC presents an improved construction of circulant sub-matrices based QC-LDPC at high SNR. The proposed construction yields a performance gain of about 1 dB at a 0.0003 bit error rate (BER). Proposed code is tested on 4 types of decoding algorithm and compared. Proposed algorithm is also compared with the existing QC-LDPC and the proposed approach outperforms the existing one at high SNR. Simulation is also performed varying the number of horizontal submatrix and the result shows that the parity check matrix with smaller horizontal concatenation shows better 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.