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arXiv:1705.09507 · IEEE Transactions on Information Theory

BP-LED Decoding Algorithm for LDPC Codes Over AWGN Channels

Irina E. Bocharova, Boris D. Kudryashov, Vitaly Skachek, Yauhen Yakimenka

Published Aug 24, 2018• 14 citations
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Abstract

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.

DOI
10.1109/tit.2018.2866879

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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. 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.
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.
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.
This paper proposes a two-stage decoding algorithm (called BP-LED) for short nonbinary low-density parity-check (LDPC) codes. It consists of the classical belief propagation (BP) decoder and a list erasure decoder (LED). Simulation results show that, for the (16, 8) LDPC code over GF(256) in the CCSDS standard, our proposed BP-LED algorithm can achieve a coding gain of 0.6 dB with respect to the FFT-QSPA. By choosing the parameters suitable for the proposed algorithm, it has a negligible performance loss with lower decoding complexity compared with the BP-MRB (most reliable basis) algorithm in [10].
Designing realizable Maximum Likelihood (ML) decoders for low-density parity-check (LDPC) codes has always been a challenging problem. Maxwell decoder is a well-known ML decoder for LDPC codes over a binary erasure channel (BEC). Based on the concept used in the Maxwell decoder, it may be possible to design near-ML decoders for LDPC codes over an additive white Gaussian noise (AWGN) channel. In this paper, we introduce a novel idea called channel transformation, which will be used as a bridge linking a BEC channel and an AWGN channel. With the transformation, we further propose a near-ML decoder for LDPC codes sent over an AWGN channel. The proposed near-ML decoder, combining a Belief Propagation (BP) decoder and a Maxwell decoder, is called the BP-Maxwell (BM) decoder. Simulation results show that compared with a BP decoder, the BM decoding algorithm can accomplish better frame error rates (FERs) by eliminating many of the error frames with a smaller percentage of error bits. Thus, the BM decoder is a step closer to the ML decoder compared with the BP decoder.