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IEEE International Symposium on Information Theory (ISIT)

Performance of ML decoding for ensembles of binary and nonbinary regular LDPC codes of finite lengths

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

Published Jun 1, 2017• 8 citations
Publisher

Abstract

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.

DOI
10.1109/isit.2017.8006637

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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.
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 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.
A universal design of low-density parity-check (LDPC) code ensembles which enables to operate reliably over various channels is of great interest. This paper considers the universality of LDPC code ensembles over sets of memoryless binary-input output-symmetric (MBIOS) channels, addressing their universality under belief-propagation (BP) and maximum-likelihood (ML) decoding. Based on the density evolution approach, closed-form, easy-to-calculate, analytical results related to the universality of LDPC code ensembles under BP decoding are derived. LP Bounds on the achievable rate of LDPC code ensembles over a set of equi-capacity channels are also derived. This paper considers the universality under ML decoding of properly selected regular LDPC code ensembles for the set of equi-capacity MBIOS channels. The full paper version related to this work [13] provides further results, full proofs, and additional discussions on the theorems.
In this paper, we derive an upper bound for the average block error probability of a standard irregular low-density parity-check (LDPC) code ensemble under the maximum-likelihood (ML) decoding. Moreover, we show that the upper bound asymptotically decreases polynomially with the code length. Furthermore, when we consider several regular LDPC code ensembles as special cases of standard irregular ones over an additive white Gaussian noise channel, we numerically show that the signal-to-noise ratio (SNR) thresholds at which the proposed bound converges to zero as the code length tends to infinity are smaller than those for a bound provided by Miller et al. We also give an example of a standard irregular LDPC code ensemble which has a lower SNR threshold than a given regular LDPC code ensemble.
Communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding is considered. The average bit error probability of an irregular LDPC code ensemble after a fixed number of iterations converges to a limit, which is calculated via density evolution, as the blocklength n tends to infinity. The difference between the bit error probability with blocklength n and the large-blocklength limit behaves asymptotically like ¿/n, where the coefficient ¿ depends on the ensemble, the number of iterations and the erasure probability of the BEC. In, ¿ is calculated for regular ensembles. In this paper, ¿ for irregular expurgated ensembles is derived. It is demonstrated that convergence of numerical estimates of ¿ to the analytic result is significantly fast for irregular unexpurgated ensembles.