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International Conference on Signal Processing

Construction of protograph LDPC codes based on Jacket matrices

Kaiyao Wang, Shaohai Hu, Yang Xiao, Moon Ho Lee, Kiseon Kim

Published Oct 1, 2010• 4 citations
Publisher

Abstract

In this paper, we propose a method to construct the protograph LDPC codes based on Jacket matrices. In the proposed code construction, the Jacket matrices are regarded as derived protograph matrices, the column weights of permutation matrices in parity check matrix depend on the elements' values of Jacket matrices. While the shift rule of the permutation matrices needs to ensure no girth 4. The proposed LDPC codes are easy to be applied for the codec of mobile digital multimedia. The simulation results in AWGN channels show that the BER performance of the proposed codes is better by about 0.5 dB than that of GB20600 LDPC codes.

DOI
10.1109/icosp.2010.5656484

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In this paper, we propose a method to construct the protograph LDPC codes based on Jacket matrices. In the proposed code construction, the Jacket matrices are regarded as derived protograph matrices, the column weights of permutation matrices in parity check matrix depend on the elements' values of Jacket matrices. While the shift rule of the permutation matrices needs to ensure no girth 4. The proposed LDPC codes are easy to be applied for the codec of mobile digital multimedia. The simulation results in AWGN channels show that the BER performance of the proposed codes is better by about 0.5 dB than that of GB20600 LDPC 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.
A novel construction method of Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) code is proposed in this paper, which is based on protograph and Balanced Block Design (BIBD) in combinational mathematics.The method is to fill the base matrix of protograph with the base set elements of BIBD in accordance with certain mathematical rules.And the base matrix of protograph proposed in the paper has special double diagonal structure, which overcomes the low weight code words in general structure and achieves rapid encoding at the same time.The QC-LDPC code in this paper possesses lower encoding complexity and less memory occupation; therefore it is easy for hardware implementation.In addition, it effectively avoids girth 4. Simulation results show that when the Bit Error Rate (BER) is 10 -7 , QC-LDPC codes based on Protograph and BIBD (PBIBD-LDPC) proposed in this paper has 0.5dB Net Coding Gain (NCG) compared with the protograph QC-LDPC codes that based on Vander monde matrix and PEG algorithm structure, and has 0.15dB NCG compared with the excellent code QC-LDPC that based on the lifting structure.
The real application of protograph LDPC codes involves two problems: fast encoding and the encoding complexity.Protograph LDPC codes suit for fast decoding comparing with other LDPC codes,but protograph LDPC codes are not necessarily able to achieve fast encoding.The encoding of existing protograph LDPC codes is done by generate matrix,but the generate matrix of protograph LDPC codes is not sparse.Thus,protograph LDPC codes exists encoding complexity,which makes it difficult for encoder hardware implementation.In order to lower the encoding complexity of protograph LDPC codes,this paper proposes a design of multi-rate protograph LDPC codes with fast encoding;The proposed multi-rate protograph LDPC codes do not exist girth-4s;Furthermore,the fast encoding algorithms of the proposed multi-rate protograph LDPC codes can reduce encoding complexity and simplify encoder hardware.The simulation results in AWGN channels show that the BER and correct error ability performance of the proposed multi-rate protograph LDPC codes is much better than that of GB20600 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.
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.