On ensembles of low-density parity-check codes: asymptotic distance distributions

Litsyn, Simon; Shevelev, Vladimir

doi:10.1109/18.992777

Cited by 147 publications

(122 citation statements)

References 12 publications

(12 reference statements)

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“…Under the optimal maximum-likelihood (ML) decoding algorithm, both the finitelength analysis and the asymptotic analysis for LDPC codes and other ensembles of turbo-like codes become tractable and rely on the weight distribution of these ensembles (see e.g. [24], [25], and [26]). Various Gallager type bounds on ML decoders for different finite LDPC code ensembles have been established in [27].…”

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confidence: 99%

Density Evolution for Asymmetric Memoryless Channels

Wang

Kulkarni²,

Poor³

2005

IEEE Trans. Inform. Theory

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Abstract-Density evolution is one of the most powerful analytical tools for low-density parity-check (LDPC) codes and graph codes with message passing decoding algorithms. With channel symmetry as one of its fundamental assumptions, density evolution (DE) has been widely and successfully applied to different channels, including binary erasure channels, binary symmetric channels, binary additive white Gaussian noise channels, etc. This paper generalizes density evolution for non-symmetric memoryless channels, which in turn broadens the applications to general memoryless channels, e.g. z-channels, composite white Gaussian noise channels, etc. The central theorem underpinning this generalization is the convergence to perfect projection for any fixed size supporting tree. A new iterative formula of the same complexity is then presented and the necessary theorems for the performance concentration theorems are developed. Several properties of the new density evolution method are explored, including stability results for general asymmetric memoryless channels. Simulations, code optimizations, and possible new applications suggested by this new density evolution method are also provided. This result is also used to prove the typicality of linear LDPC codes among the coset code ensemble when the minimum check node degree is sufficiently large. It is shown that the convergence to perfect projection is essential to the belief propagation algorithm even when only symmetric channels are considered. Hence the proof of the convergence to perfect projection serves also as a completion of the theory of classical density evolution for symmetric memoryless channels.Index Terms-Low-density parity-check (LDPC) codes, density evolution, sum-product algorithm, asymmetric channels, zchannels, rank of random matrices.

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confidence: 99%

Density Evolution for Asymmetric Memoryless Channels

Wang

Kulkarni²,

Poor³

2005

IEEE Trans. Inform. Theory

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“…Litsyn and Shevelev [68] showed that such an ensemble (referred to as "Ensemble A") has superior distance distribution compared to other ensembles they considered in [68], in the context of ML decoding.…”

Section: Remark 65mentioning

confidence: 99%

Combinatorial constructions of low-density parity check codes for iterative decoding

Vasić¹

Proceedings IEEE International Symposium on Information Theory,

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Abstract-This paper introduces several new combinatorial constructions of low-density parity-check (LDPC) codes, in contrast to the prevalent practice of using long, random-like codes. The proposed codes are well structured, and unlike random codes can lend themselves to a very low-complexity implementation. Constructions of regular Gallager codes based on cyclic difference families, cycle-invariant difference sets, and affine 1-configurations are introduced. Several constructions of difference families used for code design are presented, as well as bounds on the minimal distance of the codes based on the concept of a generalized Pasch configuration.

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“…That is for every > 0 and any sufficiently large n, there exist an LDPC code of length n and rate 1 − h(p) − that has check degree (number of 1s in a row of the parity-check matrix) at most O(log 1 ), and probability of incorrect decoding at most 2 −E L (p, )n , for some E L (p, ) > 0. We refer the reader to [6], [8] for more details of this result. Suppose we call this codeĈ.…”

Section: A Existence Of Good Codesmentioning

confidence: 99%

Local recovery properties of capacity achieving codes

Mazumdar

Chandar

Wornell

2013

2013 Information Theory and Applications Workshop (ITA)

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Abstract-A code is called locally recoverable or repairable if any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed storage where a most frequent error-event is a single storage node failure. A common objective is to repair the node by downloading data from as few other storage node as possible.In this paper we study the basic error-correcting properties of a locally recoverable code. We provide tight upper and lower bound on the local-recoverability of a code that achieves capacity of a symmetric channel. In particular it is shown that, if the coderate is less than the capacity then for the optimal codes, the maximum number of codeword symbols required to recover one lost symbol must scale as log 1 .

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On ensembles of low-density parity-check codes: asymptotic distance distributions

Cited by 147 publications

References 12 publications

Density Evolution for Asymmetric Memoryless Channels

Density Evolution for Asymmetric Memoryless Channels

Combinatorial constructions of low-density parity check codes for iterative decoding

Local recovery properties of capacity achieving codes

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