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International Conference on Wireless Communications and Signal Processing (WCSP)

A new method of detecting cycles in Tanner graph of LDPC codes

Bo Li, Gang Wang, Hongjuan Yang

Published Nov 1, 2009• 5 citations
Publisher

Abstract

To improve the performance of LDPC codes, we should detect and eliminate cycles in Tanner graph of the parity-check matrix when we make LDPC codes. We improved a logic algebraic algorithm in computing entire routes between the nodes of a communication network, and found a new method of detecting cycles in Tanner graph of LDPC codes. This method applied the logic algebraic calculation rules. First transformed the parity-check matrix and then constructed the relative matrix of the Tanner graph. Through integrating and deleting rows of relative matrix and correcting some elements, this new algorithm can detect all the cycles in a Tanner graph of LDPC codes which code length is n and information bits number is k, just needing integrating and deleting n-1 times and correcting k times. The calculation steps of the algorithm is shown detailed by an (8, 4, 2) LDPC example and its correctness have been validated.

DOI
10.1109/wcsp.2009.5371660

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To improve the performance of LDPC codes, we should detect and eliminate cycles in Tanner graph of the parity-check matrix when we make LDPC codes. We improved a logic algebraic algorithm in computing entire routes between the nodes of a communication network, and found a new method of detecting cycles in Tanner graph of LDPC codes. This method applied the logic algebraic calculation rules. First transformed the parity-check matrix and then constructed the relative matrix of the Tanner graph. Through integrating and deleting rows of relative matrix and correcting some elements, this new algorithm can detect all the cycles in a Tanner graph of LDPC codes which code length is n and information bits number is k, just needing integrating and deleting n-1 times and correcting k times. The calculation steps of the algorithm is shown detailed by an (8, 4, 2) LDPC example and its correctness have been validated.
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.
LDPC codes are typically decoded using belief propagation decoding algorithms applied to their Tanner graphs. Cycles of short lengths in these graphs, which are bipartite graphs representing parity-check equations of the codes, can negatively impact the performance of the decoder. In this paper, a trellis-based algorithm is presented to locate cycles in bipartite graphs. This algorithm is an improvement in complexity and run-time over the one developed by Lan et al. in 2004. When applied to Tanner graphs of quasi-cyclic LDPC codes, the newly proposed algorithm can exploit the structure of their parity-check matrices to cut the complexity further. Locating cycles in the Tanner graphs of codes allows for eliminating them to increase the girth and thus improve performance.
It is well known that the short cycles in Tanner Graph (TG) greatly impact on the iterative decoding performance of low density parity check (LDPC) codes. In previous literature two methods can be found for detecting such short cycles. This paper presents an improved version of previous Jun Fan's method, which is based on combinational analysis and enumeration of the check matrix patterns caused by cycles of length of 4, 6, 8 and 10 in the corresponding TG. The improved method can more exactly detect the cycles of length greater than 4 in comparison with Jun Fan's method. The validation of the proposed method can be verified by the detection results of short cycles for some LPDC codes.
In this paper, a new method is given for counting cycles in the Tanner graph of a (Type-I) quasi-cyclic (QC) low-density parity-check (LDPC) code which the complexity mainly is dependent on the base matrix, independent from the CPM-size of the constructed code. Interestingly, for large CPM-sizes, in comparison of the existing methods, this algorithm efficiently counts the cycles of any lengths in the Tanner graphs of QC-LDPC codes. In fact, the algorithm recursively counts the cycles in the paritycheck matrix column-by-column by finding all non-isomorph tailless backtrackless closed (TBC) walks in the base graph and enumerating theoretically their corresponding cycles in the same equivalent class. Moreover, this approach can be modified in few steps to find the cycle distributions of a class of LDPC codes based on Affine permutation matrices (APM-LDPC codes). Interestingly, unlike the existing methods which count the cycles up to 2g - 2, where g is the girth, the proposed algorithm can be used to enumerate the cycles of arbitrary length in the Tanner graph. Moreover, the proposed cycle searching algorithm improves upon various previously known methods, in terms of computational complexity and memory requirements.
The linear block code is called a binary Low Density Parity Check code if its parity-check matrix is a sparse matrix.For the features of parity-check matrix,a LDPC code can be illustrated by a Tanner graph.Cycles,especially short cycles in the Tanner graph,which lead to inefficient decoding and prevent the decoding algorithm from converging to the optimal decoding result.In this paper,we demonstrate that a detection method of short cycle presents generation algorithm of check matrix without cycle 4 and simulates the check-matrix.