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Journal of Beijing Jiaotong University

Construction of Quasi-Cyclic LDPC Codes Without Small Stopping Sets and Small Cycles

Yang Xiao

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The existing design of quasi-cyclic low-density parity-check(short for QC-LDPC) codes has not considered the problems of small stopping sets and small cycles together,while their existences will lead to the BER performance of QC-LDPC codes to be much poorer than that of the randomly constructed LDPC codes even decoding failure.To solve the problem,some theorems of the QC-LDPC codes without small stopping sets and small cycles were proposed which provide a theoretical basis for the construction of parity-check matrix by analysis method;meanwhile,good QC-LDPC codes without small stopping sets and small cycles were designed.Since preventing the small stopping sets and small cycles also prevents a low minimum distance,our construction method also leads to QC-LDPC codes with a higher minimum distance.The simulation results show that the QC-LDPC codes can effectively avoid specified stopping sets and small cycles and achieve good coding and decoding property due to the proposed constructions.

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International Conference on Instrumentation and Measurement, Computer, Communication and Control (IMCCC)•Jul 1, 2016•1 citation
The existing design of quasi-cyclic low-density parity-check(short for QC-LDPC) codes has not considered the problems of small stopping sets and small cycles together,while their existences will lead to the BER performance of QC-LDPC codes to be much poorer than that of the randomly constructed LDPC codes even decoding failure.To solve the problem,some theorems of the QC-LDPC codes without small stopping sets and small cycles were proposed which provide a theoretical basis for the construction of parity-check matrix by analysis method;meanwhile,good QC-LDPC codes without small stopping sets and small cycles were designed.Since preventing the small stopping sets and small cycles also prevents a low minimum distance,our construction method also leads to QC-LDPC codes with a higher minimum distance.The simulation results show that the QC-LDPC codes can effectively avoid specified stopping sets and small cycles and achieve good coding and decoding property due to the proposed constructions.
It is well known that performance of low-density parity check (LDPC) codes under iterative decoding is determined by the size of the smallest stopping sets in the Tanner graph.To solve this problem,we propose necessary and sufficient conditions of quasi-cyclic LDPC codes without small stopping sets.According to the proposed theorems and corollaries,we can design good QC-LDPC codes without small stopping sets which outperforms random LDPC codes by SNR =0.3 dB at BER of le-5,and count the number of the small stopping sets.The method can be used effectively to evaluate performance of LDPC codes according to their stopping sets distributions.It can find the number of cycles of LDPC codes which is less complex than existing algorithms.
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.
A construction method based on the p-plane to design high-girth quasi-cyclic low-density parity-check (QC-LDPC) codes is proposed. Firstly the good points in every line of the p-plane can be ascertained through filtering the bad points, because the designed parity-check matrixes using these points have the short cycles in Tanner graph of codes. Then one of the best points from the residual good points of every line in the p-plane will be found, respectively. The optimal point is also singled out according to the bit error rate (BER) performance of the QC-LDPC codes at last. Explicit necessary and sufficient conditions for the QC-LDPC codes to have no short cycles are presented which are in favor of removing the bad points in the p-plane. Since preventing the short cycles also prevents the small stopping sets, the proposed construction method also leads to QC-LDPC codes with a higher stopping distance.
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.
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.