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IEEE International Symposium on Information Theory (ISIT)

On the application of a ldpc codes to arbitrary discrete-memoryless channels

Amir Bennatan, D. Burshtein

Published Jan 1, 2003• 9 citations
Publisher
DOI
10.1109/isit.2003.1228308

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Dept. of Electrical Engineering – SystemsTel-Aviv UniversityTel-Aviv 69978, Israelabn@eng.tau.ac.il, burstyn@eng.tau.ac.ilAbstractWe examine iterative decoding of quantized coset (QC) LDPC codes, designedfor use over arbitrary channels, in particular nonbinary and asymmetric channels.An important element of QC-LDPC codes is the addition of a fixed, random vec-tor, known to the transmitter and the receiver, to all codewords. We show thatthis addition produces symmetry that significantly simplifies mathematical analy-sis. We extend several results from the analysis of LDPC codes over binary-inputsymmetric-output channels, to QC-LDPC codes with arbitrary alphabets over ar-bitrary channels, including the symmetry property and the stability condition.

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This paper first introduces an improved decoding algorithm for low-density parity-check (LDPC) codes over binary-input-output-symmetric memoryless channels. Then some fundamental properties of punctured LDPC codes are presented. It is proved that for any ensemble of LDPC codes, there exists a puncturing threshold. It is then proved that for any rates R1and R2satisfying 0121to R2resulting in asymptotically good codes for all rates R1lesRlesR2. Specifically, this implies that rates arbitrarily close to one are achievable via puncturing. Bounds on the performance of punctured LDPC codes are also presented. It is also shown that punctured LDPC codes are as good as ordinary LDPC codes. For BEC and arbitrary positive numbers R121lesRlesR2is shown. Based on the above observations, a method is proposed to design good punctured LDPC codes over a broad range of rates. Finally, it is shown that the results of this paper may be used for the proof of the existence of the capacity-achieving LDPC codes over binary-input-output-symmetric memoryless channels

IEEE Transactions on Information Theory•Jan 23, 2007•100 citations
We discuss three structures of modified low-density parity-check (LDPC) code ensembles designed for transmission over arbitrary discrete memoryless channels. The first structure is based on the well-known binary LDPC codes following constructions proposed by Gallager and McEliece, the second is based on LDPC codes of arbitrary (q-ary) alphabets employing modulo-q addition, as presented by Gallager, and the third is based on LDPC codes defined over the field GF(q). All structures are obtained by applying a quantization mapping on a coset LDPC ensemble. We present tools for the analysis of nonbinary codes and show that all configurations, under maximum-likelihood (ML) decoding, are capable of reliable communication at rates arbitrarily close to the capacity of any discrete memoryless channel. We discuss practical iterative decoding of our structures and present simulation results for the additive white Gaussian noise (AWGN) channel confirming the effectiveness of the codes.
Dept. of Electrical Engineering – SystemsTel-Aviv UniversityTel-Aviv 69978, Israelabn@eng.tau.ac.il, burstyn@eng.tau.ac.ilAbstractWe examine iterative decoding of quantized coset (QC) LDPC codes, designedfor use over arbitrary channels, in particular nonbinary and asymmetric channels.An important element of QC-LDPC codes is the addition of a fixed, random vec-tor, known to the transmitter and the receiver, to all codewords. We show thatthis addition produces symmetry that significantly simplifies mathematical analy-sis. We extend several results from the analysis of LDPC codes over binary-inputsymmetric-output channels, to QC-LDPC codes with arbitrary alphabets over ar-bitrary channels, including the symmetry property and the stability condition.
This paper first introduces an improved decoding algorithm for low-density parity-check (LDPC) codes over binary-input-output-symmetric memoryless channels. Then some fundamental properties of punctured LDPC codes are presented. It is proved that for any ensemble of LDPC codes, there exists a puncturing threshold. It is then proved that for any rates R1and R2satisfying 0121to R2resulting in asymptotically good codes for all rates R1lesRlesR2. Specifically, this implies that rates arbitrarily close to one are achievable via puncturing. Bounds on the performance of punctured LDPC codes are also presented. It is also shown that punctured LDPC codes are as good as ordinary LDPC codes. For BEC and arbitrary positive numbers R121lesRlesR2is shown. Based on the above observations, a method is proposed to design good punctured LDPC codes over a broad range of rates. Finally, it is shown that the results of this paper may be used for the proof of the existence of the capacity-achieving LDPC codes over binary-input-output-symmetric memoryless channels