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Serbian Journal of Electrical Engineering

An improved method for counting 6-cycles in low-density parity-check codes

Djamel Slimani, Abdellah Kaddai

Published Jan 1, 2023• 0 citations
Publisher

Abstract

Since their rediscovery in the early 1990s, low-density parity-check (LDPC) codes have become the most popular error-correcting codes owing to their excellent performance. An LDPC code is a linear block code that has a sparse parity-check matrix. Cycles in this matrix, particularly short cycles, degrade the performance of such a code. Hence, several methods for counting short cycles in LDPC codes have been proposed, such as Fan?s method to detect 4-cycles, 6- cycles, 8-cycles, and 10-cycles. Unfortunately, this method fails to count all 6- cycles, i.e., ignores numerous 6-cycles, in some given parity-check matrices. In this paper, an improvement of this algorithm is presented that detects all 6-cycles in LDPC codes, as well as in general bipartite graphs. Simulations confirm that the improved method offers the exact number of 6-cycles, and it succeeds in detecting those ignored by Fan?s method.

DOI
10.2298/sjee2301083s

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Since their rediscovery in the early 1990s, low-density parity-check (LDPC) codes have become the most popular error-correcting codes owing to their excellent performance. An LDPC code is a linear block code that has a sparse parity-check matrix. Cycles in this matrix, particularly short cycles, degrade the performance of such a code. Hence, several methods for counting short cycles in LDPC codes have been proposed, such as Fan?s method to detect 4-cycles, 6- cycles, 8-cycles, and 10-cycles. Unfortunately, this method fails to count all 6- cycles, i.e., ignores numerous 6-cycles, in some given parity-check matrices. In this paper, an improvement of this algorithm is presented that detects all 6-cycles in LDPC codes, as well as in general bipartite graphs. Simulations confirm that the improved method offers the exact number of 6-cycles, and it succeeds in detecting those ignored by Fan?s method.
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.
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.
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.
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.
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.