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Journal of Electrical and Computer Engineering

Design LDPC Codes without Cycles of Length 4 and 6

Jun Fan, Yang Xiao, Kiseon Kim

Published Jan 1, 2008• 22 citations
Publisher

Abstract

We present an approach for constructing LDPC codes without cycles of length 4 and 6. Firstly, we design 3 submatrices with different shifting functions given by the proposed schemes, then combine them into the matrix specified by the proposed approach, and, finally, expand the matrix into a desired parity‐check matrix using identity matrices and cyclic shift matrices of the identity matrices. The simulation result in AWGN channel verifies that the BER of the proposed code is close to those of Mackay′s random codes and Tanner′s QC codes, and the good BER performance of the proposed can remain at high code rates.

DOI
10.1155/2008/354137

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We present an approach for constructing LDPC codes without cycles of length 4 and 6. Firstly, we design 3 submatrices with different shifting functions given by the proposed schemes, then combine them into the matrix specified by the proposed approach, and, finally, expand the matrix into a desired parity‐check matrix using identity matrices and cyclic shift matrices of the identity matrices. The simulation result in AWGN channel verifies that the BER of the proposed code is close to those of Mackay′s random codes and Tanner′s QC codes, and the good BER performance of the proposed can remain at high code rates.
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.
A design algorithm of regular check matrix for Low-Density Parity-Check(LDPC)codes is proposed in this paper.The proposed approach is to design 3 sub-matrices,and each of the sub-matrices is combined by the cyclic shift matrices of the identity matrix,then to combine these sub-matrices into a desired parity check matrix according to our approach.Using the check algorithms of the short cycles to find the combinations of shifting operators by which the parity check matrix desires has no girth- four and no girth-six.The simulation results in AWGN channel show that the BER performance of these codes obtained by the generator matrix derived from the proposed check matrices to encode the random information bits is as good as that of random 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.
Low Density Parity Check (LDPC) codes are specified by the parity check matrix H. Smaller girths in the Tanner graphs of LDPC codes prevent the sum-product algorithm from converging, i.e., short cycles in H-matrix degrade the performance of LDPC decoder. In this paper, we present an algorithm for constructing LDPC codes without short cycles. The H-matrix must be sufficiently sparse to increase the girth of a code. In this work, we present a simple method to increase the sparsity of matrix and provide lemmas to avoid short cycles of lengths 4 and 6. We design sub-matrices for column-weight two and column-weight three codes. This design makes use of a simple shifting function given by the proposed scheme. These are then combined into a final H-matrix using Gray code representations. The proposed method provides structured, regular codes with flexibility in code rates and code lengths. The regularity of these codes gives the advantages like simplicity of hardware implementation and fast encoding. Key words LDPC codes, Tanner Graph, Girth, Gray codes
This paper,based on algebra and graph,proposes a method for constructing structured low-density parity-check(LDPC) codes.In this method,the authors design a connected graph with three kinds of special path to ensure that the Tanner graph of the parity check matrix mapped from the connected graph is without short cycles.The construction method results in two classes of(3,k)-regular quasi-cyclic LDPC codes with girth 8 and 12 respectively.Furthermore,by extending this construction method,two classes of LDPC codes with column weight 2,whose girths are respectively 16 and 24,can be obtained.The simulation studies show that these codes can achieve better performance than randomly constructed regular LDPC codes over AWGN channels with iterative decoding.At the BER of 10-5,the new proposed(3,6)-regular(4356,2205) QC-LDPC code performs 1.48dB from the Shannon limit.Moreover,it outperforms the random regular LDPC code of comparable parameters by 0.1dB.For the new proposed structured LDPC codes with column weight 2,they can achieve about 2dB coding gains with respect to the random regular LDPC codes with similar block lengths and code rates.
Low-density Parity-check (LDPC) coding is a class of advanced channel coding techniques. To improve the performance of LDPC codes, a simple and effective method to construct QC-LDPC codes without short cycles (cycles of length four and six) is proposed. The approach is especially effective when the parity-check matrix of QC-LDPC codes is composed of circulant permutation matrices. Firstly, we analyze the shapes of the cycles in the exponent matrix, to get the required conditions of no four-cycles and six-cycles. Then we show how to construct the parity-check matrix without four-cycles and six-cycles by filling rows of the exponent matrix with arithmetic sequences. Simulation results show that the proposed codes have better performance than array codes. Though the performance of the proposed codes is not as well as random codes at higher Eb/N0, they can be applied to practical communications because of being encoded in linear time with shift registers.