LP decoding of regular LDPC codes in memoryless channels

Halabi, Nissim; Even, Guy

doi:10.1109/isit.2010.5513612

Cited by 12 publications

(20 citation statements)

References 22 publications

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“…Unlike LP decoding where convex optimization analyses can be applied (e.g. those presented in [2][3][4]33]), ADMM-PD is based on non-convex objective functions. Therefore it is infeasible to apply convex analyses directly.…”

Section: An Instanton Analysis For Admm-pdmentioning

confidence: 99%

The ADMM Penalized Decoder for LDPC Codes

Liu

Draper

2016

IEEE Trans. Inform. Theory

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Linear programming (LP) decoding for low-density parity-check (LDPC) codes proposed by Feldman et al. is shown to have theoretical guarantees in several regimes and empirically is not observed to suffer from an error floor. However at low signal-to-noise ratios (SNRs), LP decoding is observed to have worse error performance than belief propagation (BP) decoding. In this paper, we seek to improve LP decoding at low SNRs while still achieving good high SNR performance. We first present a new decoding framework obtained by trying to solve a nonconvex optimization problem using the alternating direction method of multipliers (ADMM). This non-convex problem is constructed by adding a 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 decoder class the "ADMM penalized decoder". In our simulation results, the ADMM penalized decoder with ℓ 1 and ℓ 2 penalties outperforms both BP and LP decoding at all SNRs. For high SNR regimes where it is infeasible to simulate, we use an instanton analysis and show that the ADMM penalized decoder has better high SNR performance than BP decoding. We also develop a reweighted LP decoder using linear approximations to the objective with an ℓ 1 penalty. We show that this decoder has an improved theoretical recovery threshold compared to LP decoding. In addition, we show that the empirical gain of the reweighted LP decoder is significant at low SNRs.

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Section: An Instanton Analysis For Admm-pdmentioning

confidence: 99%

The ADMM Penalized Decoder for LDPC Codes

Liu

Draper

2016

IEEE Trans. Inform. Theory

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“…To simplify the notation, we drop iterate δ except when determining the stopping criteria. 4: repeat 5: for all i ∈ I do 6: for all j ∈ J do 9:…”

Section: Admm Penalized Decoding Algorithmmentioning

confidence: 99%

“…Compared to classic belief propagation (BP) decoding, LP decoding is far more amenable to analysis. Many decoding guarantees (e.g., [3][4][5][6][7]) have been developed for LP decoding in the literature. However, LP decoding suffers from high computational complexity when approached using generic solvers.…”

Section: Introductionmentioning

confidence: 99%

ADMM decoding of non-binary LDPC codes in F<inf>2m</inf>

Liu

Draper

2014

2014 IEEE International Symposium on Information Theory

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In this paper, we develop an efficient algorithm for linear programming (LP) decoding of non-binary low-density parity-check (LDPC) codes. We build our algorithm on the decomposition method based on the alternating direction method of multipliers (ADMM). Although expressing the LP decoding problem using ADMM is not hard, a sub-routine of ADMMprojection onto a polytope formed from embeddings of the nonbinary single parity-check code -is not straightforward. In this work, we focus on non-binary codes in fields of characteristic two. This allows us to use operations in F2 to relax the polytope under consideration into a form that is computational friendly. We introduce a rotation step that normalizes the geometry under consideration. We then apply ADMM a second time to solve the projection problem, which is a quadratic program (QP). Our decoding algorithm scales linearly with block length, linearly with check degree, and quadratically with field size.

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“…Following Koetter and Vontobel [KV06], Arora et al [ADS09] obtained improved bounds for (3, 6)-regular low-density parity-check (LDPC) codes over the binary symmetric channel (BSC). These techniques were extended to memoryless binary-input output-symmetric (MBIOS) channels [HE11] and to Tanner codes [EH11]. The proofs in these papers are based on complicated graphical structures and on a sophisticated analysis of a random min-sum process.…”

Section: Introductionmentioning

confidence: 99%

“…an LLR vector λ, then x is the unique linear-programming (LP) codeword. This proof is based on a lifting lemma [HE11] that states that local-optimality is invariant under liftings of codewords to covering graphs.…”

Section: Introductionmentioning

confidence: 99%

Local-Optimality Guarantees Based on Paths for Optimal Decoding

Even¹,

Halabi²

2013

SIAM J. Discrete Math.

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This paper presents a unified analysis framework that captures recent advances in the study of local-optimality characterizations for codes on graphs. These local-optimality characterizations are based on combinatorial structures embedded in the Tanner graph of the code. Local-optimality implies both unique maximum-likelihood (ML) optimality and unique linear-programming (LP) decoding optimality. Also, an iterative message-passing decoding algorithm is guaranteed to find the unique locally-optimal codeword, if one exists.We demonstrate this proof technique by considering a definition of local-optimality that is based on the simplest combinatorial structures in Tanner graphs, namely, paths of length h. We apply the technique of local-optimality to a family of Tanner codes. Inverse polynomial bounds in the code length are proved on the word error probability of LPdecoding for this family of Tanner codes.

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LP decoding of regular LDPC codes in memoryless channels

Cited by 12 publications

References 22 publications

The ADMM Penalized Decoder for LDPC Codes

The ADMM Penalized Decoder for LDPC Codes

ADMM decoding of non-binary LDPC codes in F<inf>2m</inf>

Local-Optimality Guarantees Based on Paths for Optimal Decoding

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