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Small Stopping Sets Counting Methods for Quasi-cyclic Low Density Parity Check Codes Based on the Cycle Theory

Yang Xiao

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
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.
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.
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.
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.