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Signal Processing

A Method of Counting the Number of Cycles in LDPC Codes

Yang Xiao

Published Jan 1, 2008• 1 citation
OpenAlex

Abstract

For a given block length sequence of the underlying Tanner graph (TG),the short circles of low-density parity check (LDPC) codes can have considerable variation in performance.By analyzing the shapes of the cycles of TG in parity check matrix,this paper presents a method of counting the number of 4-cycles,6-cycles,8-cycles and 10-cycles.Taking out a certain number of rows for different cycles,counting the number of cycles contained in these rows,then add up all the number of cycles that contained in all possi- ble combination of the rows in the matrix,and we can get the number of cycles in the matrix.This method can be used effectively to eval- uate the performance of LDPC codes according to their short circles distributions.Applying this method,we counting the number of cycles in the random LDPC codes and the quasi-cyclic LDPC codes,the BER performance shows that the random LDPC codes outperform the quasi-cyclic LDPC codes although it's girth performance is not as good as the quasi-cyclic LDPC codes.

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A Method of Counting the Number of Cycles in LDPC Codes

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For a given block length sequences of the underlying Tanner graph (TG), the short circles of low-density parity check (LDPC) codes can have considerable variation in performance. By analyzing the shapes of the cycles of TG in parity check matrix, this paper presents a method of counting the number of 4-cycles, 6-cycles, 8-cycles and 10-cycles. Taking out a certain number of rows for different cycles, counting the number of cycles contained in these rows, then adding up all the number of cycles that contained in all possible combinations of the rows in the matrix, and we can get the number of cycles in the matrix. This method can be used effectively to evaluate the performance of LDPC codes according to their short circles distributions. Applying this method, we counting the number of cycles in the random LDPC codes and the quasi-cyclic LDPC codes, the BER performance shows that the random LDPC codes outperform the quasi-cyclic LDPC codes although their girth performance is not as good as the quasi-cyclic LDPC codes.

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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.

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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.

International Conference on Instrumentation and Measurement, Computer, Communication and Control (IMCCC)•Jul 1, 2016•1 citation
For a given block length sequence of the underlying Tanner graph (TG),the short circles of low-density parity check (LDPC) codes can have considerable variation in performance.By analyzing the shapes of the cycles of TG in parity check matrix,this paper presents a method of counting the number of 4-cycles,6-cycles,8-cycles and 10-cycles.Taking out a certain number of rows for different cycles,counting the number of cycles contained in these rows,then add up all the number of cycles that contained in all possi- ble combination of the rows in the matrix,and we can get the number of cycles in the matrix.This method can be used effectively to eval- uate the performance of LDPC codes according to their short circles distributions.Applying this method,we counting the number of cycles in the random LDPC codes and the quasi-cyclic LDPC codes,the BER performance shows that the random LDPC codes outperform the quasi-cyclic LDPC codes although it's girth performance is not as good as the quasi-cyclic LDPC codes.
For a given block length sequences of the underlying Tanner graph (TG), the short circles of low-density parity check (LDPC) codes can have considerable variation in performance. By analyzing the shapes of the cycles of TG in parity check matrix, this paper presents a method of counting the number of 4-cycles, 6-cycles, 8-cycles and 10-cycles. Taking out a certain number of rows for different cycles, counting the number of cycles contained in these rows, then adding up all the number of cycles that contained in all possible combinations of the rows in the matrix, and we can get the number of cycles in the matrix. This method can be used effectively to evaluate the performance of LDPC codes according to their short circles distributions. Applying this method, we counting the number of cycles in the random LDPC codes and the quasi-cyclic LDPC codes, the BER performance shows that the random LDPC codes outperform the quasi-cyclic LDPC codes although their girth performance is not as good as the quasi-cyclic LDPC codes.
An efficient method for counting short cycles in the Tanner graphs of quasi cyclic (QC) protograph low-density parity-check (LDPC) codes is presented. The method is based on the relationship between the number of short cycles in the graph and the eigenvalues of the directed edge matrix of the graph. We demonstrate that for a QC protograph LDPC code, the complexity of computing such eigenvalues can be reduced significantly by representing the directed edge matrix as a block circulant matrix. Numerical results are presented to show the lower complexity of the proposed method compared to the existing algorithms for counting short cycles. These results also reveal that QC LDPC codes on average have a superior short cycle and girth distribution compared to similar randomly constructed codes.
It is well known that the performance of low-density parity check (LDPC) codes under iterative decoding is determined by certain combinatorial structures (such as stopping sets and girth) in the Tanner graph. The difficulty in enumerating all possible combinations of the columns may prevent an efficient search of good LDPC codes with small stopping sets and girth. To solve the problem, this paper presents a new algorithm of counting the number of small stopping sets and girth by analyzing the shapes of the cycles of Tanner graph in parity check matrix for designing good LDPC codes which is less complex than the existing algorithms. This method can be used effectively to evaluate the performance of LDPC codes according to their small stopping sets and girth distributions.
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.
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.
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.