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International Symposium on Turbo Codes and Iterative Information Processing

ML vs. BP decoding of binary and non-binary LDPC codes

Stefan Scholl, Frank Kienle, Michael Helmling, Stefan Ruzika

Published Aug 1, 2012• 8 citations
Publisher

Abstract

It has been shown that non-binary LDPC codes have a better error correction performance than binary codes for short block lengths. However, this advantage was up to now only shown under belief propagation decoding. To gain new insights, we investigate binary and non-binary codes under ML decoding. Our analysis includes different modulation schemes and decoding algorithms. For ML decoding under different modulation schemes a flexible integer programming formulation is presented. In this paper, we show that with respect to ML decoding short non-binary LDPC codes are not necessarily superior to binary codes. The decoding gain observed under BP decoding originates mainly in the more powerful non-binary decoding algorithm.

DOI
10.1109/istc.2012.6325201

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It has been shown that non-binary LDPC codes have a better error correction performance than binary codes for short block lengths. However, this advantage was up to now only shown under belief propagation decoding. To gain new insights, we investigate binary and non-binary codes under ML decoding. Our analysis includes different modulation schemes and decoding algorithms. For ML decoding under different modulation schemes a flexible integer programming formulation is presented. In this paper, we show that with respect to ML decoding short non-binary LDPC codes are not necessarily superior to binary codes. The decoding gain observed under BP decoding originates mainly in the more powerful non-binary decoding algorithm.
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.
In this paper, an iterative soft-decision hybrid decoding algorithm for non-binary low-density parity-check (LDPC) codes with short codeword lengths is proposed. The rationale of the approach is to combine the classical belief propagation (BP) iterative LDPC decoding algorithm with the most reliable basis (MRB) decoding algorithm. This allows to achieve significant performance improvements, with a complexity that, for medium/low error rates, is only slightly higher than that of the BP algorithm alone. The performance improvement with respect to pure BP decoding is up to 0.7 dB at codeword error rate (CER) ≈ 10-5. Notably, for a fixed MRB order, hybrid decoding achieves a gain up to 0.5 dB at CER ≈ 10-5with respect to BP decoding and MRB decoding used alone.
Two different decoding algorithms for non-binary LDPC codes: the fast Fourier transform belief propagation (FFT-BP) decoding algorithm and the Log-domain belief propagation (Log-BP) decoding algorithm are introduced in this paper. Their performance advantage of two different decoding algorithms for LDPC Codes over GF(q) is investigated at high code rates (1/2~1) over AWGN channel. The performance curves show that with the decreasing of code rate, both coding gains and the distances from Shannon limit increase under the two algorithms, which means for non- binary LDPC Codes to be suitable for applications at high rates. Meanwhile it is shown that Log-BP decoding algorithm may contribute a considerable saving in computational complexity at the cost of sacrifice small performance, which is important for Log-BP decoding algorithm to be applied in hardware implementation.
To solve the problem of high computational complexity and inconvenient for hardware implementation in original decoding algorithm for non-binary LDPC codes,two simplified decoding algorithms are introduced for nonbinary LDPC codes: the fast Fourier transform belief propagation(FFT-BP) decoding algorithm and the Log-domain belief propagation(Log-BP) decoding algorithm. The performance comparison between two different decoding algorithms for LDPC Codes over GF(q) is investigated at high code rates(1/2~1) over AWGN channel.The performance curves show that by the decreasing of code rate,both coding gains and the distances from Shannon limit increase with the two algorithms,which means that non-binary LDPC Codes are suit for applications at high rates.Meanwhile it is shown that Log-BP decoding algorithm may contribute a considerable improvement in computational complexity at the cost of sacrificing some performance, which is important for Log-BP decoding algorithm to be applied in hardware implementation.
Channel coding technique is a fundamental building block in any modern communication system to realize reliable, fast, and secure data transmission. At the same time, it is a challenging and crucial task, as the data transmission happens in a channel where noise, fading, and other impairments are present. The Low-Density Parity-Check (LDPC) codes give substantial results close to the Shannon limit when the complexity and processing delay time are unlimited. In this paper, the performance of the LDPC decoding with four algorithms was investigated. The investigated four algorithms were Belief Propagation (BP), Layered Belief Propagation (LBP), Normalized min-sum (NMS), and Offset min-sum (OMS). These algorithms were examined for code rates ranging from 1/3 to 9/10 and message block lengths (64, 512, 1024, and 5120) bits. The simulation results revealed the flexibility of these decoders in supporting these code rates and block lengths, which enables their usage in a wide range of applications and scenarios for fifth-generation (5G) wireless communication. In addition, the effect of the maximum number of decoding iterations on the error correction performance was investigated, and a gain of 5.6 dB can be obtained by using 32 decoding iterations at BER=2*10-3 instead of one decoding iteration. The results showed that the decoders performed better for longer message blocks than for short message blocks, and less power was required for transmitting longer messages. Finally, the comparison results of their performance in terms of bit error rate (BER) under the same conditions showed a gain of 0.8 dB using LBP at BER= 10-5 compared with the NMS decoding algorithm.
Low-density parity-check (LDPC) codes over finite fields GF(q) have a better performance than those of the binary low-density parity-check codes, at short and medium block lengths, but the decoder of GF(q)-LDPC has more complexity. In this paper, we present a new reduced-complexity belief propagation (BP) decoding algorithm for GF(q)-LDPC codes by analyzing the reliability of code symbols. Instead of update all the variable nodes during iteration, we just update variable nodes which are mostly to be in an error. The algorithm also shows a property to reduce the effects of oscillating on decoding. The simulations prove that we could improve performance and reduce complexity at the same time for short and medium GF(q)-LDPC codes.