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arXiv:1211.5757 · IEEE Transactions on Communications

Low-Complexity LP Decoding of Nonbinary Linear Codes

Mayur Punekar, Pascal O. Vontobel, Mark F. Flanagan

Published Jul 16, 2013• 16 citations
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Abstract

Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. LP decoding has been derived for binary and nonbinary linear codes. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remains prohibitively large for codes of moderate to large block length. To address this problem, two low-complexity LP (LCLP) decoding algorithms for binary linear codes have been proposed by Vontobel and Koetter, henceforth called the basic LCLP decoding algorithm and the subgradient LCLP decoding algorithm. In this paper, we generalize these LCLP decoding algorithms to nonbinary linear codes. The computational complexity per iteration of the proposed nonbinary LCLP decoding algorithms scales linearly with the block length of the code. A modified BCJR algorithm for efficient check-node calculations in the nonbinary basic LCLP decoding algorithm is also proposed, which has complexity linear in the check node degree. Several simulation results are presented for nonbinary LDPC codes defined over Z4, GF(4), and GF(8) using quaternary phase-shift keying and 8-phase-shift keying, respectively, over the AWGN channel. It is shown that for some group-structured LDPC codes, the error-correcting performance of the nonbinary LCLP decoding algorithms is similar to or better than that of the min-sum decoding algorithm.

DOI
10.1109/tcomm.2013.070213.120801

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Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. LP decoding has been derived for binary and nonbinary linear codes. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remains prohibitively large for codes of moderate to large block length. To address this problem, two low-complexity LP (LCLP) decoding algorithms for binary linear codes have been proposed by Vontobel and Koetter, henceforth called the basic LCLP decoding algorithm and the subgradient LCLP decoding algorithm. In this paper, we generalize these LCLP decoding algorithms to nonbinary linear codes. The computational complexity per iteration of the proposed nonbinary LCLP decoding algorithms scales linearly with the block length of the code. A modified BCJR algorithm for efficient check-node calculations in the nonbinary basic LCLP decoding algorithm is also proposed, which has complexity linear in the check node degree. Several simulation results are presented for nonbinary LDPC codes defined over Z4, GF(4), and GF(8) using quaternary phase-shift keying and 8-phase-shift keying, respectively, over the AWGN channel. It is shown that for some group-structured LDPC codes, the error-correcting performance of the nonbinary LCLP decoding algorithms is similar to or better than that of the min-sum decoding algorithm.
Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. The aim of LP decoding is to develop an algorithm which has error-correcting performance similar to that of the Sum-Product (SP) decoding algorithm, while at the same time it should be amenable to mathematical analysis. The LP decoding algorithm has been derived for both binary and nonbinary decoding frameworks. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remain prohibitively large for codes of moderate to large block length. To address this problem, Vontobel et al. proposed a low complexity LP decoding algorithm for binary linear codes which has complexity linear in the block length. In this paper, we extend the latter work and propose a low-complexity LP decoding algorithm for nonbinary linear codes. We use the LP formulation for the nonbinary codes as a basis and derive a pair of primal-dual LP formulations. The dual LP is then used to develop the low-complexity LP decoding algorithm for nonbinary linear codes. The complexity of the proposed algorithm is linear in the block length and is limited mainly by the maximum check node degree. As a proof of concept, we also present a simulation result for a [80, 48] LDPC code defined over ℤ4using quaternary phase-shift keying over the AWGN channel, and we show that the error-correcting performance of the proposed LP decoding algorithm is similar to that of the standard LP decoding using the simplex solver.
Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. The aim of LP decoding is to develop an algorithm which has error-correcting performance similar to that of the Sum-Product (SP) decoding algorithm, while at the same time it should be amenable to mathematical analysis. The LP decoding algorithm has also been extended to nonbinary linear codes by Flanagan et al. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remain prohibitively large for codes of moderate to large block length. To address this problem, Vontobel et al. proposed a low complexity LP decoding algorithm for binary linear codes which has complexity linear in the block length. In this paper, we extend the latter work and propose a low-complexity LP decoding algorithm for nonbinary linear codes. We use the LP formulation proposed by Flanagan et al. as a basis and derive a pair of primal-dual LP formulations. The dual LP is then used to develop the low-complexity LP decoding algorithm for nonbinary linear codes. In contrast to the binary low-complexity LP decoding algorithm, our proposed algorithm is not directly related to the nonbinary SP algorithm. Nevertheless, the complexity of the proposed algorithm is linear in the block length and is limited mainly by the maximum check node degree. As a proof of concept, we also present a simulation result for a [80,48][80,48][80,48] LDPC code defined over Z4\mathbb{Z}_4Z4​ using quaternary phase-shift keying over the AWGN channel, and we show that the error-correcting performance of the proposed LP decoding algorithm is similar to that of the standard LP decoding using the simplex solver.
Linear Programming (LP) decoding is emerging as an attractive alternative to decode Low-Density Parity-Check (LDPC) codes. However, the earliest LP decoders proposed for binary and nonbinary LDPC codes are not suitable for use at moderate and large code lengths. To overcome this problem, Vontobel et al. developed an iterative Low-Complexity LP (LCLP) decoding algorithm for binary LDPC codes. The variable and check node calculations of binary LCLP decoding algorithm are related to those of binary Belief Propagation (BP). The present authors generalized this work to derive an iterative LCLP decoding algorithm for nonbinary linear codes. Contrary to binary LCLP, the variable and check node calculations of this algorithm are in general different from that of nonbinary BP. The overall complexity of nonbinary LCLP decoding is linear in block length; however the complexity of its check node calculations is exponential in the check node degree. In this paper, we propose a modified BCJR algorithm for efficient check node processing in the nonbinary LCLP decoding algorithm. The proposed algorithm has complexity linear in the check node degree. We also introduce an alternative state metric to improve the run time of the proposed algorithm. Simulation results are presented for (504, 252) and (1008, 504) nonbinary LDPC codes over ℤ4.
The problem of low-complexity linear programming (LP) decoding of nonbinary low-density parity-check (LDPC) codes is considered, and an iterative LP decoding algorithm is presented. Results that were previously derived for binary LDPC codes are extended to the nonbinary case. Both simple and generalized nonbinary LDPC codes are considered. It is shown how the algorithm can be implemented efficiently using a finite-field fast Fourier transform. Then, the convergence rate of the algorithm is analyzed. The complexity of the algorithm scales linearly in the block length, and it can approximate, up to an arbitrarily small relative error, the objective function of the exact LP solution. When applied to a typical code from an appropriate nonbinary LDPC code ensemble, the algorithm can correct a constant fraction of errors in linear (in the block length) computational complexity. Computer experiments with the new iterative LP decoding algorithm show that, in the error floor region, it can have better performance compared to belief propagation decoding, with similar computational requirements.
Linear Programming (LP) decoding is emerging as an attractive alternative to decode Low-Density Parity-Check (LDPC) codes. However, the earliest LP decoders proposed for binary and nonbinary LDPC codes are not suitable for use at moderate and large code lengths. To overcome this problem, Vontobel et al. developed an iterative Low-Complexity LP (LCLP) decoding algorithm for binary LDPC codes. The variable and check node calculations of binary LCLP decoding algorithm are related to those of binary Belief Propagation (BP). The present authors generalized this work to derive an iterative LCLP decoding algorithm for nonbinary linear codes. Contrary to binary LCLP, the variable and check node calculations of this algorithm are in general different from that of nonbinary BP. The overall complexity of nonbinary LCLP decoding is linear in block length; however the complexity of its check node calculations is exponential in the check node degree. In this paper, we propose a modified BCJR algorithm for efficient check node processing in the nonbinary LCLP decoding algorithm. The proposed algorithm has complexity linear in the check node degree. We also introduce an alternative state metric to improve the run time of the proposed algorithm. Simulation results are presented for (504,252)(504, 252)(504,252) and (1008,504)(1008, 504)(1008,504) nonbinary LDPC codes over Z4\mathbb{Z}_4Z4​.
The problem of low complexity linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered. An iterative algorithm, similar to min-sum and belief propagation, for efficient approximate solution of this problem was proposed by Vontobel and Koetter. In this paper, the convergence rate and computational complexity of this algorithm are studied using a scheduling scheme that we propose. In particular, we are interested in obtaining a feasible vector in the LP decoding problem that is close to optimal in the following sense. The distance, normalized by the block length, between the minimum and the objective function value of this approximate solution can be made arbitrarily small. It is shown that such a feasible vector can be obtained with a computational complexity which scales linearly with the block length. Combined with previous results that have shown that the LP decoder can correct some fixed fraction of errors we conclude that this error correction can be achieved with linear computational complexity. This is achieved by first applying the iterative LP decoder that decodes the correct transmitted codeword up to an arbitrarily small fraction of erroneous bits, and then correcting the remaining errors using some standard method. These conclusions are also extended to generalized LDPC codes.