Improved Bounds on the Word Error Probability of&amp;lt;tex&amp;gt;$RA(2)$&amp;lt;/tex&amp;gt;Codes With Linear-Programming-Based Decoding

Halabi, Nissim; Even, Guy

doi:10.1109/tit.2004.839509

Cited by 14 publications

(14 citation statements)

References 7 publications

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“…We further note that the improvement over Feldman's work presented in [6] for q = 2 does not seem to extend to q > 2. This is due to the following.…”

Section: Discussion and Numerical Resultscontrasting

confidence: 53%

“…a set of atom paths, possibly with multiple copies of the same atom path in the set. The set Ψ is also 6 required to satisfy a certain "agreeability" constraint. Formally, define, for each segment i in the trellis where 1 ≤ i ≤ n, the following multiset B i :…”

Section: An Error Bound For Regular Ra(q) Codes With Even Qmentioning

confidence: 99%

“…In [13], regular RA(2) codes were examined, based on flow theory and graph-theoretical arguments. Halabi and Even [6] have proposed a better bound on RA (2) codes which is based on a more careful examination of the underlying graph-theoretical nature of the problem. Irregular RA code ensembles have been shown to achieve excellent performance under iterative message-passing decoding.…”

mentioning

confidence: 99%

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Error bounds for repeat-accumulate codes decoded via linear programming

Goldenberg

Burshtein

2010

2010 6th International Symposium on Turbo Codes &Amp; Iterative Information Processing

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We examine regular and irregular repeat-accumulate (RA) codes with repetition degrees which are all even. For these codes and with a particular choice of an interleaver, we give an upper bound on the decoding error probability of a linear-programming based decoder which is an inverse polynomial in the block length. Our bound is valid for any memoryless, binary-input, output-symmetric (MBIOS) channel. This result generalizes the bound derived by Feldman et al., which was for regular RA(2) codes.

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“…We further note that the improvement over Feldman's work presented in [6] for q = 2 does not seem to extend to q > 2. This is due to the following.…”

Section: Discussion and Numerical Resultscontrasting

confidence: 53%

Section: An Error Bound For Regular Ra(q) Codes With Even Qmentioning

confidence: 99%

mentioning

confidence: 99%

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Error bounds for repeat-accumulate codes decoded via linear programming

Goldenberg

Burshtein

2010

2010 6th International Symposium on Turbo Codes &Amp; Iterative Information Processing

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“…The technique of LP decoding was introduced for turbo-like codes [10], extended to LDPC codes [11], [14], and further studied by various researchers (e.g., [28], [5], [12], [8], [13], [16]). Significant recent interest has focused on postprocessing algorithms that use the ML-certificate property of LP decoding to achieve near ML performance (see [8], [4]) and also [9], [26]).…”

Section: A Previous Workmentioning

confidence: 99%

Probabilistic Analysis of Linear Programming Decoding

Daskalakis

Dimakis

Karp

et al. 2008

IEEE Trans. Inform. Theory

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Abstract-We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous nonasymptotic results for LDPC codes, and in particular, exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a flow on the Tanner graph of the code. An interesting by-product of our analysis is to establish the existence of "probabilistic expansion" in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst case setting.

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“…Every codeword is a vertex of the relaxed LP polytope, however, usually there are additional vertices for which at least one component is non-integral. LP decoding has been applied to several codes, among them: cycle codes, turbo-like and RA codes [FK04,HE05,GB11], LDPC codes [FMS + 07, DDKW08, KV06, ADS09, HE11], and expander codes [FS05,Ska11]. Our work is motivated by the problem of finite-length and average-case analysis of successful LP decoding of binary Tanner codes.…”

Section: Introductionmentioning

confidence: 99%

On Decoding Irregular Tanner Codes With Local-Optimality Guarantees

Halabi

Even

2014

IEEE Trans. Inform. Theory

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We consider decoding of binary linear Tanner codes using message-passing iterative decoding and linear programming (LP) decoding in memoryless binary-input output symmetric (MBIOS) channels. We present new certificates that are based on a combinatorial characterization for local-optimality of a codeword in irregular Tanner codes with respect to any MBIOS channel. This characterization is a generalization of [Arora, Daskalakis, Steurer, Proc. ACM Symp. Theory of Computing, 2009] and [Vontobel, Proc. Inf. Theory and Appl. Workshop, 2010] and is based on a conical combination of normalized weighted subtrees in the computation trees of the Tanner graph. These subtrees may have any finite height h (even equal or greater than half of the girth of the Tanner graph). In addition, the degrees of local-code nodes in these subtrees are not restricted to two (i.e., these subtrees are not restricted to skinny trees). We prove that local optimality in this new characterization implies maximum-likelihood (ML) optimality and LP optimality, and show that a certificate can be computed efficiently.We also present a new message-passing iterative decoding algorithm, called normalized weighted min-sum (NWMS). NWMS decoding is a belief-propagation (BP) type algorithm that applies to any irregular binary Tanner code with single parity-check local codes (e.g., LDPC codes and HDPC codes). We prove that if a locally-optimal codeword with respect to height parameter h exists (whereby notably h is not limited by the girth of the Tanner graph), then NWMS decoding finds this codeword in h iterations. The decoding guarantee of the NWMS decoding algorithm applies whenever there exists a locally optimal codeword. Because local optimality of a codeword implies that it is the unique ML codeword, the decoding guarantee also provides an ML certificate for this codeword.Finally, we apply the new local optimality characterization to regular Tanner codes, and prove lower bounds on the noise thresholds of LP decoding in MBIOS channels. When the noise is below these lower bounds, the probability that LP decoding fails to decode the transmitted codeword decays doubly exponentially in the girth of the Tanner graph.

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Improved Bounds on the Word Error Probability of<tex>$RA(2)$</tex>Codes With Linear-Programming-Based Decoding

Cited by 14 publications

References 7 publications

Error bounds for repeat-accumulate codes decoded via linear programming

Error bounds for repeat-accumulate codes decoded via linear programming

Probabilistic Analysis of Linear Programming Decoding

On Decoding Irregular Tanner Codes With Local-Optimality Guarantees

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