Multiple-Bases Belief-Propagation Decoding of High-Density Cyclic Codes
Thorsten Hehn, Johannes B. Huber, Olgica Milenkovic, Stefan Laendner
备注：
This full paper accompanies a letter submitted to "IEEE Transactions on Communications". It is intended to provide detailed information for interested readers of the letter. 24 pages, 6 figures
We introduce a new method for decoding short and moderate length linear block codes with dense parity-check matrix representations of cyclic form, termed multiple-bases belief-propagation (MBBP). The proposed iterative scheme makes use of the fact that a code has many structurally diverse parity-check matrices, capable of detecting different error patterns. We show that this inherent code property leads to decoding algorithms with significantly better performance when compared to standard BP decoding. Furthermore, we describe how to choose sets of parity-check matrices of cyclic form amenable for multiple-bases decoding, based on analytical studies performed for the binary erasure channel. For several cyclic and extended cyclic codes, the MBBP decoding performance can be shown to closely follow that of maximum-likelihood decoders.
It is shown how to combine ordered statistics decoding (OSD) with CRC-aided belief propagation list (CBPL) decoding of polar codes. Even when the reprocessing order of the OSD is as low as one, the new decoder is shown to significantly improve on CBPL. For reprocessing orders higher than one, we suggest partial reprocessing, where only error patterns associated with the least reliable part of the belief propagation decoded most reliable independent bits are considered. This type of partial reprocessing offers a trade-off between performance and computational complexity.
This paper presents a novel propagation (BP) based decoding algorithm for polar codes. The proposed algorithm facilitates belief propagation by utilizing the specific constituent codes that exist in the factor graph, which results in an express journey (XJ) for belief information to propagate in each decoding iteration. In addition, this XJ-BP decoder employs a novel round-trip message passing scheduling method for the increased efficiency. The proposed method simplifies min-sum (MS) BP decoder by 40.6%. Along with the round-trip scheduling, the XJ-BP algorithm reduces the computational comple…
Permutation decoding gained recent interest as it can exploit the symmetries of a code in a parallel fashion. Moreover, it has been shown that by viewing permuted polar codes as polar subcodes, the set of usable permutations in permutation decoding can be increased. We extend this idea to pre-transformed polar codes, such as cyclic redundancy check (CRC)-aided polar codes, which previously could not be decoded using permutations due to their lack of automorphisms. Using belief propagation (BP)-based subdecoders, we showcase a performance close to CRC-aided SCL (CA-SCL) decoding. The proposed a…
This paper presents an enhanced belief propagation (BP) decoding algorithm and a reinforcement learning-based BP decoding algorithm for polar codes. The enhanced BP algorithm weighs each Processing Element (PE) input based on their signals and Euclidean distances using a heuristic metric. The proposed reinforcement learning-based BP decoding strategy relies on reweighting the messages and consists of two steps: we first weight each PE input based on their signals and Euclidean distances using a heuristic metric, then a Q-learning algorithm (QLBP) is employed to figure out the best correction f…
This paper defines cyclic and minimal derivative descendants (DDs) of an extended cyclic code from the derivative of the Mattson-Solomon polynomials, respectively. First, it demonstrates that the cyclic DDs are the same extended cyclic code. It allows us to perform soft-decision decoding for extended cyclic codes based on their cyclic DDs. Then, it proves that the minimal DDs are equivalent codes. It also allows us to perform soft-decision decoding based on the minimal DDs with permutations. Simulation results show that our proposed derivative decoding can be close to the maximum likelihood de…
We consider the effect of LLR saturation on belief propagation decoding of low-density parity-check codes. Saturation occurs universally in practice and is known to have a significant effect on error floor performance. Our focus is on threshold analysis and stability of density evolution. We analyze the decoder for certain low-density parity-check code ensembles and show that belief propagation decoding generally degrades gracefully with saturation. Stability of density evolution is, on the other hand, rather strongly affected by saturation and the asymptotic qualitative effect of saturation i…
The decoding performance of polar codes strongly depends on the decoding algorithm used, while also the decoder throughput and its latency mainly depend on the decoding algorithm. In this work, we implement the powerful successive cancellation list (SCL) decoder on a GPU and identify the bottlenecks of this algorithm with respect to parallel computing and its difficulties. The inherent serial decoding property of the SCL algorithm naturally limits the achievable speed-up gains on GPUs when compared to CPU implementations. In order to increase the decoding throughput, we use a hybrid decoding s…
In this paper a new message passing algorithm, which takes advantage of both tree-based re-parameterization and the knowledge of short cycles, is introduced for the purpose of decoding LDPC codes with short block lengths. The proposed algorithm is called variable factor appearance probability belief propagation (VFAP-BP) algorithm and is suitable for wireless communications applications, where both good decoding performance and low-latency are expected. Our simulation results show that the VFAP-BP algorithm outperforms the standard BP algorithm and requires a significantly smaller number of it…
A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding thr…
We consider the effect of log-likelihood ratio saturation on belief propagation decoder low-density parity-check codes. Saturation is commonly done in practice and is known to have a significant effect on error floor performance. Our focus is on threshold analysis and stability of density evolution. We analyze the decoder for standard low-density parity-check code ensembles and show that belief propagation decoding generally degrades gracefully with saturation. Stability of density evolution is, on the other hand, rather strongly effected by saturation and the asymptotic qualitative effect of …
Short survey about code on the graph by example of hardware friendly quasi-cycle LDPC code. We consider two main properties of code: weight enumerator (well known from classic code theory) and Trapping sets pseudocodewords weight spectrum (a subgraph in code graph's which become reasone of decoding failure under Belief propagation). In paper we show fast methods to measure first components of TS enumerator using EMD=ACE constrains for high girth codes on the graph using graph spectral classification method. This approach simplify solving trouble of agreed between minimal TS pseudocode weight a…
Although iterative decoding of polar codes has recently made huge progress based on the idea of permuted factor graphs, it still suffers from a non-negligible performance degradation when compared to state-of-the-art CRC-aided successive cancellation list (CA-SCL) decoding. In this work, we show that iterative decoding of polar codes based on the belief propagation list (BPL) algorithm can approach the error-rate performance of CA-SCL decoding and, thus, can be efficiently used for decoding the standardized 5G polar codes. Rather than only utilizing the cyclic redundancy check (CRC) as a stopp…
A sequential updating scheme (SUS) for belief propagation (BP) decoding of LDPC codes over Galois fields, $GF(q)$, and correlated Markov sources is proposed, and compared with the standard parallel updating scheme (PUS). A thorough experimental study of various transmission settings indicates that the convergence rate, in iterations, of the BP algorithm (and subsequently its complexity) for the SUS is about one half of that for the PUS, independent of the finite field size $q$. Moreover, this 1/2 factor appears regardless of the correlations of the source and the channel's noise model, while t…
In this paper, we propose a class of finite alphabet iterative decoder (FAID), called mutual information-maximizing quantized belief propagation (MIM-QBP) decoder, for decoding regular low-density parity-check (LDPC) codes. Our decoder follows the reconstruction-calculation-quantization (RCQ) decoding architecture that is widely used in FAIDs. We present the first complete and systematic design framework for the RCQ parameters, and prove that our design with sufficient precision at node update is able to maximize the mutual information between coded bits and exchanged messages. Simulation resu…
We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterativ…
New algorithms for efficient decoding of polar codes (which may be CRC-augmented), transmitted over either a binary erasure channel (BEC) or an additive white Gaussian noise channel (AWGNC), are presented. We start by presenting a new efficient exact maximum likelihood decoding algorithm for the BEC based on inactivation decoding and analyze its computational complexity. This algorithm applies a matrix triangulation process on a sparse polar code parity check matrix, followed by solving a small size linear system over GF(2). We then consider efficient decoding of polar codes, transmitted over …
In this paper, we present a belief propagation (BP) based algorithm for decoding non-orthogonal space-time block codes (STBC) from cyclic division algebras (CDA) having {\em large dimensions}. The proposed approach involves message passing on Markov random field (MRF) representation of the STBC MIMO system. Adoption of BP approach to decode non-orthogonal STBCs of large dimensions has not been reported so far. Our simulation results show that the proposed BP-based decoding achieves increasingly closer to SISO AWGN performance for increased number of dimensions. In addition, it also achieves ne…
A new lower bound on the minimum Hamming distance of linear quasi-cyclic codes over finite fields is proposed. It is based on spectral analysis and generalizes the Semenov- Trifonov bound in a similar way as the Hartmann-Tzeng bound extends the BCH approach for cyclic codes. Furthermore, a syndrome-based algebraic decoding algorithm is given.
Owing to their capacity-achieving performance and low encoding and decoding complexity, polar codes have drawn much research interests recently. Successive cancellation decoding (SCD) and belief propagation decoding (BPD) are two common approaches for decoding polar codes. SCD is sequential in nature while BPD can run in parallel. Thus BPD is more attractive for low latency applications. However BPD has some performance degradation at higher SNR when compared with SCD. Concatenating LDPC with Polar codes is one popular approach to enhance the performance of BPD , where a short LDPC code is use…
In this work, we show that polar belief propagation (BP) decoding exhibits an error floor behavior which is caused by clipping of the log-likelihood ratios (LLR). The error floor becomes more pronounced for clipping to smaller LLR-values. We introduce a single-value measure quantifying a "relative error floor", showing, by exhaustive simulations for different lengths, that the error floor is mainly caused by inadequate clipping values. We propose four modifications to the conventional BP decoding algorithm to mitigate this error floor behavior, demonstrating that the error floor is a decoder p…
Belief propagation (BP) decoding of low-density parity-check (LDPC) codes with various dynamic decoding schedules have been proposed to improve the efficiency of the conventional flooding schedule. As the ultimate goal of an ideal LDPC code decoder is to have correct bit decisions, a dynamic decoding schedule should be variable node (VN)-centric and be able to find the VNs with probable incorrect decisions and having a good chance to be corrected if chosen for update. We propose a novel and effective metric called conditional innovation (CI) which serves this design goal well. To make the most…
Polar codes are newly discovered capacity-achieving codes, which have attracted lots of research efforts. Polar codes can be efficiently decoded by the low-complexity successive cancelation (SC) algorithm and the SC list (SCL) decoding algorithm. The belief propagation (BP) decoding algorithm not only is an alternative to the SC and SCL decoders, but also provides soft outputs that are necessary for joint detection and decoding. Both the BP decoder and the soft cancelation (SCAN) decoder were proposed for polar codes to output soft information about the coded bits. In this paper, first a belie…
We describe a novel approach to interpret a polar code as a low-density parity-check (LDPC)-like code with an underlying sparse decoding graph. This sparse graph is based on the encoding factor graph of polar codes and is suitable for conventional belief propagation (BP) decoding. We discuss several pruning techniques based on the check node decoder (CND) and variable node decoder (VND) update equations, significantly reducing the size (i.e., decoding complexity) of the parity-check matrix. As a result, iterative polar decoding can then be conducted on a sparse graph, akin to the traditional w…
This paper proposes a belief propagation (BP) message passing algorithm based joint multiple symbol differential detection (MSDD) and channel decoding scheme for noncoherent differential ultra-wideband impulse radio (UWB-IR) systems. MSDD is an effective means to improve the performance of noncoherent differential UWB-IR systems. To optimize the overall detection and decoding performance, in this paper, we propose a novel soft-in soft-out (SISO) MSDD scheme and its integration with SISO channel decoding for noncoherent differential UWB-IR. we first propose a new auto-correlation receiver (AcR)…
In practice, LDPC codes are decoded using message passing methods. These methods offer good performance but tend to converge slowly and sometimes fail to converge and to decode the desired codewords correctly. Recently, tree-reweighted message passing methods have been modified to improve the convergence speed at little or no additional complexity cost. This paper extends this line of work and proposes a new class of locally optimized reweighting strategies, which are suitable for both regular and irregular LDPC codes. The proposed decoding algorithm first splits the factor graph into subgraph…
In this paper, we show that Quasi-Cyclic LDPC codes can efficiently accommodate the hybrid iterative/ML decoding over the binary erasure channel. We demonstrate that the quasi-cyclic structure of the parity-check matrix can be advantageously used in order to significantly reduce the complexity of the ML decoding. This is achieved by a simple row/column permutation that transforms a QC matrix into a pseudo-band form. Based on this approach, we propose a class of QC-LDPC codes with almost ideal error correction performance under the ML decoding, while the required number of row/symbol operations…
A list decoding algorithm for matrix-product codes is provided when $C_1,..., C_s$ are nested linear codes and $A$ is a non-singular by columns matrix. We estimate the probability of getting more than one codeword as output when the constituent codes are Reed-Solomon codes. We extend this list decoding algorithm for matrix-product codes with polynomial units, which are quasi-cyclic codes. Furthermore, it allows us to consider unique decoding for matrix-product codes with polynomial units.
We propose a belief propagation list (BPL) decoder with comparable performance to the successive cancellation list (SCL) decoder of polar codes, which already achieves the maximum likelihood (ML) bound of polar codes for sufficiently large list size $L$. The proposed decoder is composed of multiple parallel independent belief propagation (BP) decoders based on differently permuted polar code factor graphs. A list of possible transmitted codewords is generated and the one closest to the received vector, in terms of Euclidean distance, is picked. To the best of our knowledge, the proposed BPL de…
In this paper, we study the impact of locality on the decoding of binary cyclic codes under two approaches, namely ordered statistics decoding (OSD) and trellis decoding. Given a binary cyclic code having locality or availability, we suitably modify the OSD to obtain gains in terms of the Signal-To-Noise ratio, for a given reliability and essentially the same level of decoder complexity. With regard to trellis decoding, we show that careful introduction of locality results in the creation of cyclic subcodes having lower maximum state complexity. We also present a simple upper-bounding techniqu…
For finite length polar codes, channel polarization leaves a significant number of channels not fully polarized. Adding a Cyclic Redundancy Check (CRC) to better protect information on the semi-polarized channels has already been successfully applied in the literature, and is straightforward to be used in combination with Successive Cancellation List (SCL) decoding. Belief Propagation (BP) decoding, however, offers more potential for exploiting parallelism in hardware implementation, and thus, we focus our attention on improving the BP decoder. Specifically, similar to the CRC strategy in the …
We consider near maximum-likelihood (ML) decoding of short linear block codes. In particular, we propose a novel decoding approach based on neural belief propagation (NBP) decoding recently introduced by Nachmani et al. in which we allow a different parity-check matrix in each iteration of the algorithm. The key idea is to consider NBP decoding over an overcomplete parity-check matrix and use the weights of NBP as a measure of the importance of the check nodes (CNs) to decoding. The unimportant CNs are then pruned. In contrast to NBP, which performs decoding on a given fixed parity-check matri…
We consider automorphism ensemble decoding (AED) of quasi-cyclic (QC) low-density parity-check (LDPC) codes. Belief propagation (BP) decoding on the conventional factor graph is equivariant to the quasi-cyclic automorphisms and therefore prevents gains by AED. However, by applying small modifications to the parity-check matrix at the receiver side, we can break the symmetry without changing the code at the transmitter. This way, we can leverage a gain in error-correcting performance using an ensemble of identical BP decoders, without increasing the worst-case decoding latency. The proposed met…
We show that the performance of iterative belief propagation (BP) decoding of polar codes can be enhanced by decoding over different carefully chosen factor graph realizations. With a genie-aided stopping condition, it can achieve the successive cancellation list (SCL) decoding performance which has already been shown to achieve the maximum likelihood (ML) bound provided that the list size is sufficiently large. The proposed decoder is based on different realizations of the polar code factor graph with randomly permuted stages during decoding. Additionally, a different way of visualizing the p…
The recent success in constructing asymptotically good quantum low-density parity-check (QLDPC) codes makes this family of codes a promising candidate for error-correcting schemes in quantum computing. However, conventional belief propagation (BP) decoding of QLDPC codes does not yield satisfying performance due to the presence of unavoidable short cycles in their Tanner graph and the special degeneracy phenomenon. In this work, we propose to decode QLDPC codes based on a check matrix with redundant rows, generated from linear combinations of the rows in the original check matrix. This approac…
The decoding of Low-Density Parity-Check codes by the Belief Propagation (BP) algorithm is revisited. We check the iterative algorithm for its convergence to a codeword (termination), we run Monte Carlo simulations to find the probability distribution function of the termination time, n_it. Tested on an example [155, 64, 20] code, this termination curve shows a maximum and an extended algebraic tail at the highest values of n_it. Aiming to reduce the tail of the termination curve we consider a family of iterative algorithms modifying the standard BP by means of a simple relaxation. The relaxat…
Polar codes have promising error-correction capabilities. Yet, decoding polar codes is often challenging, particularly with large blocks, with recently proposed decoders based on list-decoding or neural-decoding. The former applies multiple decoders or the same decoder multiple times with some redundancy, while the latter family utilizes emerging deep learning schemes to learn to decode from data. In this work we introduce a novel polar decoder that combines the list-decoding with neural-decoding, by forming an ensemble of multiple weighted belief-propagation (WBP) decoders, each trained to de…
Recently, error correcting codes in the erasure channel have drawn great attention for various applications such as distributed storage systems and wireless sensor networks, but many of their decoding algorithms are not practical because they have higher decoding complexity and longer delay. Thus, the automorphism group decoder (AGD) for cyclic codes in the erasure channel was introduced, which has good erasure decoding performance with low decoding complexity. In this paper, we propose new two-stage AGDs (TS-AGDs) for cyclic codes in the erasure channel by modifying the parity check matrix an…
The standard algebraic decoding algorithm of cyclic codes $[n,k,d]$ up to the BCH bound $t$ is very efficient and practical for relatively small $n$ while it becomes unpractical for large $n$ as its computational complexity is $O(nt)$. Aim of this paper is to show how to make this algebraic decoding computationally more efficient: in the case of binary codes, for example, the complexity of the syndrome computation drops from $O(nt)$ to $O(t\sqrt n)$, and that of the error location from $O(nt)$ to at most $\max \{O(t\sqrt n), O(t^2\log(t)\log(n))\}$.
A new lower bound on the minimum distance of q-ary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are special cases of our bound. For some classes of codes the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean Algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula.
Algebraic codes such as BCH code are receiving renewed interest as their short block lengths and low/no error floors make them attractive for ultra-reliable low-latency communications (URLLC) in 5G wireless networks. This paper aims at enhancing the traditional adaptive belief propagation (ABP) decoding, which is a soft-in-soft-out (SISO) decoding for high-density parity-check (HDPC) algebraic codes, such as Reed-Solomon (RS) codes, Bose-Chaudhuri-Hocquenghem (BCH) codes, and product codes. The key idea of traditional ABP is to sparsify certain columns of the parity-check matrix corresponding …
We consider spectrally-efficient communication over a Rayleigh N-block-fading channel with a K- sparse L-length discrete-time impulse response (for 0<K<L<N), where neither the transmitter nor receiver know the channel's coefficients nor its support. Since the high-SNR ergodic capacity of this channel has been shown to obey C(SNR) = (1-K/N)log2(SNR)+O(1), any pilot-aided scheme that sacrifices more than K dimensions per fading block to pilots will be spectrally inefficient. This causes concern about the conventional "compressed channel sensing" approach, which uses O(K polylog L) pilots. In thi…
In this paper, we present an iterative soft-decision decoding algorithm for Reed-Solomon codes offering both complexity and performance advantages over previously known decoding algorithms. Our algorithm is a list decoding algorithm which combines two powerful soft decision decoding techniques which were previously regarded in the literature as competitive, namely, the Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation based on adaptive parity check matrices, recently proposed by Jiang and Narayanan. Building on the Jiang-Narayanan algorithm, we present a belief-pr…
Belief propagation is a powerful tool in statistical physics, machine learning, and modern coding theory. As a decoding method, it is ubiquitous in classical error correction and has also been applied to stabilizer-based quantum error correction. The algorithm works by passing messages between nodes of the factor graph associated with the code and enables efficient decoding, in some cases even up to the Shannon capacity of the channel. Here we construct a belief propagation algorithm which passes quantum messages on the factor graph and is capable of decoding the classical-quantum channel with…
A belief-propagation decoder for low-density lattice codes is given which represents messages explicitly as a mixture of Gaussians functions. The key component is an algorithm for approximating a mixture of several Gaussians with another mixture with a smaller number of Gaussians. This Gaussian mixture reduction algorithm iteratively reduces the number of Gaussians by minimizing the distance between the original mixture and an approximation with one fewer Gaussians. Error rates and noise thresholds of this decoder are compared with those for the previously-proposed decoder which discretely qua…
Quantum error correction is necessary to protect logical quantum states and operations. However, no meaningful data protection can be made when the syndrome extraction is erroneous due to faulty measurement gates. Quantum data-syndrome (DS) codes are designed to protect the data qubits and syndrome bits concurrently. In this paper, we propose an efficient decoding algorithm for quantum DS codes with sparse check matrices. Based on a refined belief propagation (BP) decoding for stabilizer codes, we propose a DS-BP algorithm to handle the quaternary quantum data errors and binary syndrome bit er…
Near optimal decoding of good error control codes is generally a difficult task. However, for a certain type of (sufficiently) good codes an efficient decoding algorithm with near optimal performance exists. These codes are defined via a combination of constituent codes with low complexity trellis representations. Their decoding algorithm is an instance of (loopy) belief propagation and is based on an iterative transfer of constituent beliefs. The beliefs are thereby given by the symbol probabilities computed in the constituent trellises. Even though weak constituent codes are employed close t…
Codes based on sparse matrices have good performance and can be efficiently decoded by belief-propagation (BP). Decoding binary stabilizer codes needs a quaternary BP for (additive) codes over GF(4), which has a higher check-node complexity compared to a binary BP for codes over GF(2). Moreover, BP decoding of stabilizer codes suffers a performance loss from the short cycles in the underlying Tanner graph. In this paper, we propose a refined BP algorithm for decoding quantum codes by passing scalar messages. For a given error syndrome, this algorithm decodes to the same output as the conventio…
The cyclically equivariant neural decoder was recently proposed in [Chen-Ye, International Conference on Machine Learning, 2021] to decode cyclic codes. In the same paper, a list decoding procedure was also introduced for two widely used classes of cyclic codes -- BCH codes and punctured Reed-Muller (RM) codes. While the list decoding procedure significantly improves the Frame Error Rate (FER) of the cyclically equivariant neural decoder, the Bit Error Rate (BER) of the list decoding procedure is even worse than the unique decoding algorithm when the list size is small. In this paper, we propo…
In this paper, we propose a non-binary belief propagation approach (NB-BP) for detection of $M$-ary modulation symbols and decoding of $q$-ary LDPC codes in large-scale multiuser MIMO systems. We first propose a message passing based symbol detection algorithm which computes vector messages using a scalar Gaussian approximation of interference, which results in a total complexity of just $O(KN\sqrt{M})$, where $K$ is the number of uplink users and $N$ is the number of base station (BS) antennas. The proposed NB-BP detector does not need to do a matrix inversion, which gives a complexity advant…
