Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1102.3902 · IEEE International Symposium on Information Theory (ISIT)

Polytope of correct (linear programming) decoding and low-weight pseudo-codewords

Michael Chertkov, Mikhail Stepanov

Published Jul 1, 2011• 10 citations
arXivHTMLPDF

Abstract

We analyze Linear Programming (LP) decoding of graphical binary codes operating over soft-output, symmetric and log-concave channels. We show that the error-surface, separating domain of the correct decoding from domain of the erroneous decoding, is a polytope. We formulate the problem of finding the lowest-weight pseudo-codeword as a non-convex optimization (maximization of a convex function) over a polytope, with the cost function defined by the channel and the polytope defined by the structure of the code. This formulation suggests new provably convergent heuristics for finding the lowest weight pseudo-codewords improving in quality upon previously discussed. The algorithm performance is tested on the example of the Tanner [155,64,20] code over the Additive White Gaussian Noise (AWGN) channel.

DOI
10.1109/isit.2011.6033824

Related papers

An Efficient Pseudocodeword Search Algorithm for Linear Programming Decoding of LDPC Codes

Michael Chertkov, Mikhail Stepanov

In linear programming (LP) decoding of a low-density parity-check (LDPC) code one minimizes a linear functional, with coefficients related to log-likelihood ratios, over a relaxation of the polytope spanned by the codewords. In order to quantify LP decoding it is important to study vertexes of the relaxed polytope, so-called pseudocodewords. We propose a technique to heuristcally create a list of pseudocodewords close to the zero codeword and their distances. Our pseudocodeword-search algorithm starts by randomly choosing configuration of the noise. The configuration is modified through a discrete number of steps. Each step consists of two substeps: one applies an LP decoder to the noise-configuration deriving a pseudocodeword, and then finds configuration of the noise equidistant from the pseudocodeword and the zero codeword. The resulting noise configuration is used as an entry for the next step. The iterations converge rapidly to a pseudocodeword neighboring the zero codeword. Repeated many times, this procedure is characterized by the distribution function of the pseudocodeword effective distance. The efficiency of the procedure is demonstrated on examples of the Tanner code and Margulis codes operating over an additive white Gaussian noise (AWGN) channel.

arXiv:cs/0601113•IEEE Transactions on Information Theory•Mar 27, 2008•56 citations

Looping linear programming decoding of LDPC codes

Michael Chertkov, Mikhail Stepanov

Abstract — Considering the Linear Programming (LP) decoding of Low-Density-Parity-Check (LDPC) code [1], we describe an efficient algorithm for finding pseudo codewords with the smallest effective distance. The algorithm, coined LP-loop, starts from choosing randomly initial configuration of the noise. The configuration is modified through a discrete number of steps. Each step consists of two sub-steps. First, one applies LP decoder to the initial noise-configuration deriving a pseudo-codeword. Second, one finds configuration of the noise equidistant from the pseudo codeword and the all zeros codeword. The resulting noise configuration is used as an entry for the next step of the LP-loop algorithm. The iterations converge fast to a pseudo codeword neighboring the all zeros codeword — domain of the pseudo codeword attraction shares a common piece of error-surface with the all zeros codeword domain. Repeated many times the LP-loop procedure is characterized by the distribution function (frequency spectrum) of the pseudo codeword effective distance. Effective distance of the coding scheme is approximated by the shortest distance pseudo-codeword in the spectrum. Efficiency of the LP-loop algorithm is demonstrated on examples of the Tanner (155, 64, 20) code and Margulis p = 7 and p = 11 codes (672 and 2640 bits long respectively) performing over Additive-White-Gaussian-Noise (AWGN) channel.

•Jan 27, 2006•2 citations

Linear-Programming Decoding of Nonbinary Linear Codes

Mark F. Flanagan, Vitaly Skachek, Eimear Byrne +1 more

A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates LP-based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the ldquomaximum-likelihood (ML) certificaterdquo property. It is also shown that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. It is also proved that if the modulator-channel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. Two alternative polytopes for use with LP decoding are studied, and it is shown that for many classes of codes these polytopes yield a complexity advantage for decoding. These polytope representations lead to polynomial-time decoders for a wide variety of classical nonbinary linear codes. LP decoding performance is illustrated for ternary Golay code with ternary phase-shift keying (PSK) modulation over additive white Gaussian noise (AWGN), and in this case it is shown that the performance of the LP decoder is comparable to codeword-error-rate-optimum hard-decision-based decoding. LP decoding is also simulated for medium-length ternary and quaternary low-density parity-check (LDPC) codes with corresponding PSK modulations over AWGN.

arXiv:0707.4360•IEEE Transactions on Information Theory•Aug 24, 2009•66 citations

On adaptive linear programming decoding of linear codes over GF(8)

Eirik Rosnes, Michael Helmling

In this work, we consider adaptive linear programming (LP) decoding of linear codes over GF(8). In particular, we give explicit constructions of valid inequalities (using no auxiliary variables) for the codeword polytope (or the convex hull) of the so-called constant-weight embedding of a single parity-check code over GF(8) that all are facet-defining. We conjecture that these inequalities together with so-called simplex constraints give a complete and irredundant description of the embedded (under the constant-weight embedding) codeword polytope. Furthermore, these sets of inequalities are used to develop an efficient (as compared to a static approach) exact (assuming that the conjecture is true) adaptive LP decoder for linear codes over GF(8). Numerical results show that only a very small subset of these inequalities is necessary for achieving close-to-exact LP decoding performance.

Information Theory and Applications Workshop (ITA)•Jan 1, 2016•1 citation

Pseudo-Codeword Analysis of Tanner Graphs From Projective and Euclidean Planes

Roxana Smarandache, Pascal O. Vontobel

We consider coded data transmission over a binary-input output-symmetric memoryless channel using a binary linear code. In order to understand the performance of maximum-likelihood (ML) decoding, one studies the codewords, in particular the minimal codewords, and their Hamming weights. In the context of linear programming (LP) decoding, one's attention needs to be shifted to the pseudo-codewords, in particular, to the minimal pseudo-codewords and their pseudo-weights. In this paper, we investigate some families of codes that have good properties under LP decoding, namely certain families of low-density parity-check (LDPC) codes that are derived from projective and Euclidean planes: we study the structure of their minimal pseudo-codewords and give lower bounds on their pseudo-weight. Besides this main focus, we also present some results that hold for pseudo-codewords and minimal pseudo-codewords of any Tanner graph, and we highlight how the importance of minimal pseudo-codewords under LP decoding varies depending on which binary-input output-symmetric memoryless channel is used.

arXiv:cs/0602089•IEEE Transactions on Information Theory•Jun 28, 2007•50 citations

Guessing Facets: Polytope Structure and Improved LP Decoder

Alexandros G. Dimakis, Martin J. Wainwright

A new approach for decoding binary linear codes by solving a linear program (LP) over a relaxed codeword polytope was recently proposed by Feldman et al. In this paper we investigate the structure of the polytope used in the LP relaxation decoding. We begin by showing that for expander codes, every fractional pseudocodeword always has at least a constant fraction of non-integral bits. We then prove that for expander codes, the active set of any fractional pseudocodeword is smaller by a constant fraction than the active set of any codeword. We exploit this fact to devise a decoding algorithm that provably outperforms the LP decoder for finite blocklengths. It proceeds by guessing facets of the polytope, and resolving the linear program on these facets. While the LP decoder succeeds only if the ML codeword has the highest likelihood over all pseudocodewords, we prove that for expander codes the proposed algorithm succeeds even with a constant number of pseudocodewords of higher likelihood. Moreover, the complexity of the proposed algorithm is only a constant factor larger than that of the LP decoder

IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2006•24 citations
We analyze Linear Programming (LP) decoding of graphical binary codes operating over soft-output, symmetric and log-concave channels. We show that the error-surface, separating domain of the correct decoding from domain of the erroneous decoding, is a polytope. We formulate the problem of finding the lowest-weight pseudo-codeword as a non-convex optimization (maximization of a convex function) over a polytope, with the cost function defined by the channel and the polytope defined by the structure of the code. This formulation suggests new provably convergent heuristics for finding the lowest weight pseudo-codewords improving in quality upon previously discussed. The algorithm performance is tested on the example of the Tanner [155,64,20] code over the Additive White Gaussian Noise (AWGN) channel.
In linear programming (LP) decoding of a low-density parity-check (LDPC) code one minimizes a linear functional, with coefficients related to log-likelihood ratios, over a relaxation of the polytope spanned by the codewords. In order to quantify LP decoding it is important to study vertexes of the relaxed polytope, so-called pseudocodewords. We propose a technique to heuristcally create a list of pseudocodewords close to the zero codeword and their distances. Our pseudocodeword-search algorithm starts by randomly choosing configuration of the noise. The configuration is modified through a discrete number of steps. Each step consists of two substeps: one applies an LP decoder to the noise-configuration deriving a pseudocodeword, and then finds configuration of the noise equidistant from the pseudocodeword and the zero codeword. The resulting noise configuration is used as an entry for the next step. The iterations converge rapidly to a pseudocodeword neighboring the zero codeword. Repeated many times, this procedure is characterized by the distribution function of the pseudocodeword effective distance. The efficiency of the procedure is demonstrated on examples of the Tanner code and Margulis codes operating over an additive white Gaussian noise (AWGN) channel.
Abstract — Considering the Linear Programming (LP) decoding of Low-Density-Parity-Check (LDPC) code [1], we describe an efficient algorithm for finding pseudo codewords with the smallest effective distance. The algorithm, coined LP-loop, starts from choosing randomly initial configuration of the noise. The configuration is modified through a discrete number of steps. Each step consists of two sub-steps. First, one applies LP decoder to the initial noise-configuration deriving a pseudo-codeword. Second, one finds configuration of the noise equidistant from the pseudo codeword and the all zeros codeword. The resulting noise configuration is used as an entry for the next step of the LP-loop algorithm. The iterations converge fast to a pseudo codeword neighboring the all zeros codeword — domain of the pseudo codeword attraction shares a common piece of error-surface with the all zeros codeword domain. Repeated many times the LP-loop procedure is characterized by the distribution function (frequency spectrum) of the pseudo codeword effective distance. Effective distance of the coding scheme is approximated by the shortest distance pseudo-codeword in the spectrum. Efficiency of the LP-loop algorithm is demonstrated on examples of the Tanner (155, 64, 20) code and Margulis p = 7 and p = 11 codes (672 and 2640 bits long respectively) performing over Additive-White-Gaussian-Noise (AWGN) channel.
A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates LP-based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the ldquomaximum-likelihood (ML) certificaterdquo property. It is also shown that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. It is also proved that if the modulator-channel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. Two alternative polytopes for use with LP decoding are studied, and it is shown that for many classes of codes these polytopes yield a complexity advantage for decoding. These polytope representations lead to polynomial-time decoders for a wide variety of classical nonbinary linear codes. LP decoding performance is illustrated for ternary Golay code with ternary phase-shift keying (PSK) modulation over additive white Gaussian noise (AWGN), and in this case it is shown that the performance of the LP decoder is comparable to codeword-error-rate-optimum hard-decision-based decoding. LP decoding is also simulated for medium-length ternary and quaternary low-density parity-check (LDPC) codes with corresponding PSK modulations over AWGN.
In this work, we consider adaptive linear programming (LP) decoding of linear codes over GF(8). In particular, we give explicit constructions of valid inequalities (using no auxiliary variables) for the codeword polytope (or the convex hull) of the so-called constant-weight embedding of a single parity-check code over GF(8) that all are facet-defining. We conjecture that these inequalities together with so-called simplex constraints give a complete and irredundant description of the embedded (under the constant-weight embedding) codeword polytope. Furthermore, these sets of inequalities are used to develop an efficient (as compared to a static approach) exact (assuming that the conjecture is true) adaptive LP decoder for linear codes over GF(8). Numerical results show that only a very small subset of these inequalities is necessary for achieving close-to-exact LP decoding performance.
We consider coded data transmission over a binary-input output-symmetric memoryless channel using a binary linear code. In order to understand the performance of maximum-likelihood (ML) decoding, one studies the codewords, in particular the minimal codewords, and their Hamming weights. In the context of linear programming (LP) decoding, one's attention needs to be shifted to the pseudo-codewords, in particular, to the minimal pseudo-codewords and their pseudo-weights. In this paper, we investigate some families of codes that have good properties under LP decoding, namely certain families of low-density parity-check (LDPC) codes that are derived from projective and Euclidean planes: we study the structure of their minimal pseudo-codewords and give lower bounds on their pseudo-weight. Besides this main focus, we also present some results that hold for pseudo-codewords and minimal pseudo-codewords of any Tanner graph, and we highlight how the importance of minimal pseudo-codewords under LP decoding varies depending on which binary-input output-symmetric memoryless channel is used.
A new approach for decoding binary linear codes by solving a linear program (LP) over a relaxed codeword polytope was recently proposed by Feldman et al. In this paper we investigate the structure of the polytope used in the LP relaxation decoding. We begin by showing that for expander codes, every fractional pseudocodeword always has at least a constant fraction of non-integral bits. We then prove that for expander codes, the active set of any fractional pseudocodeword is smaller by a constant fraction than the active set of any codeword. We exploit this fact to devise a decoding algorithm that provably outperforms the LP decoder for finite blocklengths. It proceeds by guessing facets of the polytope, and resolving the linear program on these facets. While the LP decoder succeeds only if the ML codeword has the highest likelihood over all pseudocodewords, we prove that for expander codes the proposed algorithm succeeds even with a constant number of pseudocodewords of higher likelihood. Moreover, the complexity of the proposed algorithm is only a constant factor larger than that of the LP decoder