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International Symposium on Turbo Codes and Iterative Information Processing

Wrap-around sliding-window near-ML decoding of binary LDPC codes over the BEC

Irina E. Bocharova, Boris D. Kudryashov, Eirik Rosnes, Vitaly Skachek, Øyvind Ytrehus

Published Sep 1, 2016• 2 citations
Publisher

Abstract

A novel method of low-complexity near-maximum-likelihood (ML) decoding of quasi-cyclic (QC) low-density parity-check (LDPC) codes over the binary erasure channel is presented. The idea is similar to wrap-around decoding of tail-biting convolutional codes. ML decoding is applied to a relatively short window which is cyclically shifted along the received sequence. The procedure is repeated until either all erasures have been corrected, or no new erasures are corrected at a certain round. A new upper bound on the ensemble-average ML decoding error probability for a finite-length row-regular LDPC code family is derived and presented. Furthermore, a few examples of regular and irregular QC LDPC codes are studied by simulations and their performance is compared with the ensemble-average performance. Finally, the impact of the codeword weight and stopping set size spectra on the ML and belief-propagation decoding performance is discussed.

DOI
10.1109/istc.2016.7593068

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A novel method of low-complexity near-maximum-likelihood (ML) decoding of quasi-cyclic (QC) low-density parity-check (LDPC) codes over the binary erasure channel is presented. The idea is similar to wrap-around decoding of tail-biting convolutional codes. ML decoding is applied to a relatively short window which is cyclically shifted along the received sequence. The procedure is repeated until either all erasures have been corrected, or no new erasures are corrected at a certain round. A new upper bound on the ensemble-average ML decoding error probability for a finite-length row-regular LDPC code family is derived and presented. Furthermore, a few examples of regular and irregular QC LDPC codes are studied by simulations and their performance is compared with the ensemble-average performance. Finally, the impact of the codeword weight and stopping set size spectra on the ML and belief-propagation decoding performance is discussed.
The performance of maximum-likelihood (ML) decoding on the binary erasure channel for finite-length low-density parity-check (LDPC) codes from two random ensembles is studied. The theoretical average spectrum of the Gallager ensemble is computed by using a recurrent procedure and compared to the empirically found average spectrum for the same ensemble as well as to the empirical average spectrum of the Richardson-Urbanke ensemble and spectra of selected codes from both ensembles. Distance properties of the random codes from the Gallager ensemble are discussed. A tightened union-type upper bound on the ML decoding error probability based on the precise coefficients of the average spectrum is presented. A new upper bound on the ML decoding performance of LDPC codes from the Gallager ensemble based on computing the rank of submatrices of the code parity-check matrix is derived. A new low-complexity near-ML decoding algorithm for quasi-cyclic LDPC codes is proposed and simulated. Its performance is compared to the upper bounds on the ML decoding performance.
A new method is presented for low-complexity near-maximum-likelihood (ML) decoding of low-density parity-check (LDPC) codes over the additive white Gaussian noise channel. The proposed method termed belief-propagation-list erasure decoding (BP-LED) is based on erasing carefully chosen unreliable bits performed in case of BP decoding failure. A strategy of introducing erasures into the received vector and a new erasure decoding algorithm are proposed. The new erasure decoding algorithm, called list erasure decoding, combines ML decoding over the BEC with list decoding applied if the ML decoder fails to find a unique solution. The asymptotic exponent of the average list size for random regular LDPC codes from the Gallager ensemble is analyzed. Furthermore, a few examples of irregular quasi-cyclic LDPC as well as randomly constructed regular LDPC codes of short and moderate lengths are studied by simulations and their performance is compared to the tightened upper bound on the LDPC ensemble-average performance and the upper bound on the average performance of random linear codes under ML decoding. A comparison of the BP decoding and BP-LED performance of the WiMAX standard codes and performance of the near-ML BEAST decoding are presented. The new algorithm is applied to decoding a short nonbinary (NB) LDPC code over extensions of the binary Galois field. The obtained simulation results are compared to the tightened upper bound on the ensemble-average performance of the binary image of regular NB LDPC codes.
The performance of maximum-likelihood (ML) decoding on the binary erasure channel for finite-length low-density parity-check (LDPC) codes from two random ensembles is studied. A tightened union-type upper bound on the ML decoding error probability based on the precise coefficients of the average weight spectrum is presented. For LDPC codes from the Gallager ensemble and the Richardson-Urbanke ensemble, new upper bounds on the ML decoding performance based on computing the rank of submatrices of the code parity-check matrix are derived. A new lower bound on the ML decoding threshold followed from the latter error probability bound is obtained. An improved lower bound on the error probability for codes with a known estimate on the minimum distance is presented as well. A new low-complexity near-ML decoding algorithm for quasi-cyclic LDPC codes is proposed and simulated. Its performance is compared to the simulated belief propagation and ML decoding performance and simulated performance of the best known improved iterative decoding techniques, as well as, with the derived upper bounds on the ML decoding performance and with decoding thresholds obtained by the density evolution technique.
In this paper, we are concerned with the finite-length analysis of low-density parity-check (LDPC) codes when used over the binary erasure channel (BEC). The main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of LDPC codes when decoded iteratively. We also give expressions for upper bounds on the average bit and block erasure probability for regular LDPC ensembles and the standard random ensemble under maximum-likelihood (ML) decoding. Finally, we present what we consider to be the most important open problems in this area.
This paper investigates decoding of low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We study the iterative and maximum-likelihood (ML) decoding of LDPC codes on this channel. We derive bounds on the ML decoding of LDPC codes on the BEC. We then present an improved decoding algorithm. The proposed algorithm has almost the same complexity as the standard iterative decoding. However, it has better performance. Simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm. We also provide some graph-theoretic properties of different decoding algorithms of LDPC codes over the BEC which we think are useful to better understand the LDPC decoding methods, in particular, for finite-length codes.
The Gallager ensembles of binary regular LDPC codes and binary images of nonbinary regular LDPC codes are studied. Recurrent procedure for computing average spectra for these two ensembles is presented. By using the existing bounding techniques, estimates on the error probability of the maximum-likelihood (ML) decoding over an AWGN channel with BPSK signaling for short codes from different ensembles of LDPC codes are obtained. The numerical results show performance of the ML decoding for different code ensembles. Conclusions drawn based on the average code spectra are then verified by near-ML decoding simulations for both randomly selected and the best known short codes. The asymptotic ML decoding thresholds for AWGN and BSC channels are calculated. As expected, codes with the ML decoding performance superior to that of the average code in the ensemble, are easy to find. However, comparison of the the presented results with simulation results for belief propagation (BP) decoding shows that the ML decoding performance should not be used as a target for searching for good iteratively decodable codes.