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IEEE Transactions on Information Theory

Iterative Linear Programming Decoding of Nonbinary LDPC Codes With Linear Complexity

Dina Goldin, David Burshtein

Published Aug 6, 2012• 17 citations
Publisher

Abstract

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.

DOI
10.1109/tit.2012.2211859

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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.
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.
The problem of low complexity linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered. An iterative algorithm 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. In particular we are interested in obtaining a feasible vector in the LP decoding problem, with objective function value whose distance to the minimum, normalized by the block length, can be made arbitrarily small. It is shown that such a feasible vector can be obtained in linear, in the block length, computational complexity. 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.
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.
The iterative algorithm, for low complexity linear programming (LP) decoding of low-density parity-check (LDPC) codes, proposed by Vontobel and Koetter, is considered. 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, with objective function value whose distance to the minimum, normalized by the block length, can be made arbitrarily small. It is shown that such a feasible vector can be obtained with linear, in the block length, computational complexity. Improved bounds on the convergence rate are also presented. The results extend to generalized LDPC (GLDPC) codes. It is also shown that previous results for LDPC and GLDPC codes, on the ability of the LP decoder to correct some fixed fraction of errors, hold with linear computational complexity when using the approximate iterative LP decoder.
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.
The problem of low-complexity approximated linear programming (LP) decoding of LDPC codes over GF(q) is considered. An iterative algorithm with linear complexity was proposed by Burshtein for the binary case. However, this algorithm cannot be trivially generalized for the non-binary case, since the derivation uses the even parity property of the check nodes, which does not hold for codes over GF(q). In this work the algorithm is generalized to the non-binary case. We show, that by applying this algorithm to a softened version of the non-binary LP problem proposed by Flanagan, Skachek, Byrne and Grefeath, we can obtain, with complexity linear in the block length, a feasible solution vector for the non-binary LP decoding problem, which is arbitrarily close to optimal in the following sense. The distance between the minimum value achieved by the exact LP decoder and the objective function value of the approximate solution, normalized by the block length of the code, can be made arbitrarily small. We present simulation results and comparisons with the belief propagation (BP) algorithm using-regular ternary LDPC codes. The proposed algorithm can be easily extended to generalized LDPC codes.