A Separation Algorithm for Improved LP-Decoding of Linear Block Codes

Tanatmis, Akin; Ruzika, Stefan; Hamacher, Horst W.; Punekar, Mayur; Kienle, Frank; Wehn, Norbert

doi:10.1109/tit.2010.2048489

Cited by 39 publications

(44 citation statements)

References 22 publications

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“…The goal is to produce a modified LP that eliminates the (formerly) optimum pseudo-codeword without eliminating any binary vertexes, hopefully yielding the ML solution. A number of proposals add additional linear constraints, e.g., redundant parity-checks (RPCs) [3], [4], [5], [6], or "lift and project" [2]. An alternate approach is to add a small number of integer (actually binary) constraints [7], [8], giving a mixed integer linear program (MILP).…”

Section: Introductionmentioning

confidence: 99%

Multi-stage decoding of LDPC codes

Wang

Yedidia

Draper

2009

2009 IEEE International Symposium on Information Theory

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In this paper we present a three-stage decoding strategy that combines quantized and unquantized belief propagation (BP) decoders with a mixed-integer linear programming (MILP) decoder. Each decoding stage is activated only when the preceeding stage fails to converge to a valid codeword. The faster BP decoding stages are able to correct most errors, yielding a short average decoding time. Only in the rare cases when the iterative stages fail is the slower but more powerful MILP decoder used. The MILP decoder iteratively adds binary constraints until either the maximum likelihood codeword is found or some maximum number of binary constraints has been added. Simulation results demonstrate a large improvement in the word error rate (WER) of the proposed multi-stage decoder in comparison to belief propagation. The improvement is particularly noticeable in the low crossover probability (error floor) regime. Through introduction of an accelerated "active-set" version of the quantized BP decoder we significantly speed up the pace of simulation to simulate low density parity check (LDPC) codes of length up to around 2000 down to a WER of around 10(10) on the binary symmetric channel. We demonstrate that for certain codes our approach can efficiently approach the optimal ML decoding performance for low crossover probabilities. IEEE International Symposium on Information TheoryThis work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Abstract-In this paper we present a three-stage decoding strategy that combines quantized and un-quantized belief propagation (BP) decoders with a mixed-integer linear programming (MILP) decoder. Each decoding stage is activated only when the preceding stage fails to converge to a valid codeword. The faster BP decoding stages are able to correct most errors, yielding a short average decoding time. Only in the rare cases when the iterative stages fail is the slower but more powerful MILP decoder used. The MILP decoder iteratively adds binary constraints until either the maximum likelihood codeword is found or some maximum number of binary constraints has been added.Simulation results demonstrate a large improvement in the word error rate (WER) of the proposed multi-stage decoder in comparison to belief propagation. The improvement is particularly noticeable in the low crossover probability (error floor) regime. Through introduction of an accelerated "active-set" version of ...

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Section: Introductionmentioning

confidence: 99%

Multi-stage decoding of LDPC codes

Wang

Yedidia

Draper

2009

2009 IEEE International Symposium on Information Theory

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“…A separation algorithm (SA) for improving the errorcorrecting performance of LP decoding was introduced in 2010 by Tanatmis et al [15], which is abbreviated as SALP in the following contents. Thereafter, Zhang and Siegel [16] provided a novel decoder that combines the new adaptive cutgenerating (ACG) algorithm with the ALP algorithm, called ACG-ALP decoder.…”

Section: Linear Programming Decodingmentioning

confidence: 99%

“…As an approximation to ML decoding, Linear programming (LP) decoding was first proposed by Feldman et al [13]. Other LP-based decoding algorithms are developed to improve the speed or performance of Feldman et al's LP decoder [14], [15], [16], [17]. Although most of LP decoding algorithms are utilized to decode LDPC codes, they also worked very well when utilized to decode algebraic block codes, such as BCH codes, Golay codes, and the (89, 45, 17) QR code, e.g., see [15], [17], [11].…”

Section: Introductionmentioning

confidence: 99%

Algebraic and linear programming decoding of the (73, 37, 13) quadratic residue code

Liu

Chen

et al. 2014

2014 IEEE International Conference on Communications (ICC)

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In this paper 1 , a method to search the subsets I and J needed in computing the unknown syndromes for the (73, 37, 13) quadratic residue (QR) code is proposed. According to the resulting I and J, one computes the unknown syndromes, and thus finds the corresponding error-locator polynomial by using an inverse-free BM algorithm. Based on the modified Chase-II algorithm, the performance of soft-decision decoding for the (73, 37, 13) QR code is given. This result is never seen in the literature, to our knowledge. Moreover, the error-rate performance of linear programming (LP) decoding for the (73, 37, 13) QR code is also investigated, and LP-based decoding is shown to be significantly superior in performance to the algebraic soft-decision decoding while requiring almost the same computational complexity.Index Terms-Berlekamp-Massey algorithm, quadratic residue code, Chase algorithm, linear programming.

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“…3: Initialize λ j as the all-zeros vector and z j as the all 0.5 vector for all j ∈ J . 4: repeat 5: for all i ∈ I do 6:…”

Section: A Using 1 Penalty For Decodingmentioning

confidence: 99%

“…These works can roughly be classified into two categories: (a) change the constraint set of LP decoding by tightening the relaxation, e.g. [6], [7] or (b) change the objective functions, e.g. [8].…”

Section: Introductionmentioning

confidence: 99%

The l<inf>1</inf> penalized decoder and its reweighted LP

Liu

Draper

Recht

2012

2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Linear programming (LP) decoding for low density parity check (LDPC) codes proposed by Feldman et al. has attracted considerable attention in recent years. Despite having theoretical guarantees in some regimes, at low SNRs LP decoding is observed to have worse error performance than belief propagation (BP) decoding. In this paper, we present a novel decoding algorithm obtained by trying to solve a non-convex optimization problem using the alternating direction method of multipliers (ADMM). This non-convex problem is constructed by adding an 1 penalty term to the LP decoding objective. The goal of the penalty term is to make "pseudocodewords", which are the non-integer vertices of the LP relaxation to which the LP decoder fails, more costly. We name this the " 1 penalized decoder". We also develop a reweighted LP decoding algorithm base on this 1 penalized objective. We show that this reweighted LP has an improved theoretical recovery threshold compared to the original LP. In addition, simulations show that, in comparison to LP decoding, these two decoders both achieve significantly lower error rates and are not observed to have an "error floor". In particular, the 1 penalized decoder meets or outperforms the BP decoder at all SNRs.

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A Separation Algorithm for Improved LP-Decoding of Linear Block Codes

Cited by 39 publications

References 22 publications

Multi-stage decoding of LDPC codes

Multi-stage decoding of LDPC codes

Algebraic and linear programming decoding of the (73, 37, 13) quadratic residue code

The l<inf>1</inf> penalized decoder and its reweighted LP

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