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1. A Reconstruction-Computation-Quantization (RCQ) Approach to Node Operations in LDPC Decoding

   https://doi.org/10.1109/GLOBECOM42002.2020.9348139  

   Wang, Linfang; Wesel, Richard D.; Stark, Maximilian; Bauch, Gerhard (December 2020, GLOBECOM 2020 - 2020 IEEE Global Communications Conference)

   null (Ed.)

   This paper proposes a finite-precision decoding method for low-density parity-check (LDPC) codes that features the three steps of Reconstruction, Computation, and Quantization (RCQ). Unlike Mutual-Information-Maximization Quantized Belief Propagation (MIM-QBP), RCQ can approximate either belief propagation or Min-Sum decoding. MIM-QBP decoders do not work well when the fraction of degree-2 variable nodes is large. However, sometimes a large fraction of degree-2 variable nodes is used to facilitate a fast encoding structure, as seen in the IEEE 802.11 standard and the DVB-S2 standard. In contrast to MIM-QBP, the proposed RCQ decoder may be applied to any off-the-shelf LDPC code, including those with a large fraction of degree-2 variable nodes. Simulations show that a 4-bit Min-Sum RCQ decoder delivers frame error rate (FER) performance within 0.1 dB of floating point belief propagation (BP) for the IEEE 802.11 standard LDPC code in the low SNR region. The RCQ decoder actually outperforms floating point BP and Min-Sum in the high SNR region were FER less than 10 −5 . This paper also introduces Hierarchical Dynamic Quantization (HDQ) to design the time-varying non-uniform quantizers required by RCQ decoders. HDQ is a low-complexity design technique that is slightly sub-optimal. Simulation results comparing HDQ and optimal quantization on the symmetric binary-input memoryless additive white Gaussian noise channel show a mutual information loss of less than 10 −6 bits, which is negligible in practice. 

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2. FPGA Implementations of Layered MinSum LDPC Decoders Using RCQ Message Passing

   https://doi.org/10.1109/GLOBECOM46510.2021.9685732  

   Terrill, Caleb; Wang, Linfang; Chen, Sean; Hulse, Chester; Kuo, Calvin; Wesel, Richard; Divsalar, Dariush (December 2021, IEEE)

   Non-uniform message quantization techniques such as reconstruction-computation-quantization (RCQ) improve error-correction performance and decrease hardware complexity of low-density parity-check (LDPC) decoders that use a flooding schedule. Layered MinSum RCQ (L-msRCQ) enables message quantization to be utilized for layered decoders and irregular LDPC codes. We investigate field-programmable gate array (FPGA) implementations of L-msRCQ decoders. Three design methods for message quantization are presented, which we name the Lookup, Broadcast, and Dribble methods. The decoding performance and hardware complexity of these schemes are compared to a layered offset MinSum (OMS) decoder. Simulation results on a (16384, 8192) protograph-based raptor-like (PBRL) LDPC code show that a 4-bit L-msRCQ decoder using the Broadcast method can achieve a 0.03 dB improvement in error-correction performance while using 12% fewer registers than the OMS decoder. A Broadcast-based 3-bit L-msRCQ decoder uses 15% fewer lookup tables, 18% fewer registers, and 13% fewer routed nets than the OMS decoder, but results in a 0.09 dB loss in performance. 

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3. Neural Polar Decoders for DNA Data Storage

   https://doi.org/10.1109/JSAIT.2025.3610751  

   Aharoni, Ziv; Pfister, Henry D (January 2025, IEEE Journal on Selected Areas in Information Theory)

   Synchronization errors, arising from both synthesis and sequencing noise, present a fundamental challenge in DNA-based data storage systems. These errors are often modeled as insertion-deletion-substitution (IDS) channels, for which maximum-likelihood decoding is quite computationally expensive. In this work, we propose a data-driven approach based on neural polar decoders (NPDs) to design decoders with reduced complexity for channels with synchronization errors. The proposed architecture enables decoding over IDS channels with reduced complexity O(AN log N ), where A is a tunable parameter independent of the channel. NPDs require only sample access to the channel and can be trained without an explicit channel model. Additionally, NPDs provide mutual information (MI) estimates that can be used to optimize input distributions and code design. We demonstrate the effectiveness of NPDs on both synthetic deletion and IDS channels. For deletion channels, we show that NPDs achieve near-optimal decoding performance and accurate MI estimation, with significantly lower complexity than trellis-based decoders. We also provide numerical estimates of the channel capacity for the deletion channel. We extend our evaluation to realistic DNA storage settings, including channels with multiple noisy reads and real-world Nanopore sequencing data. Our results show that NPDs match or surpass the performance of existing methods while using significantly fewer parameters than the state-of-the-art. These findings highlight the promise of NPDs for robust and efficient decoding in DNA data storage systems. 

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4. A Threshold-Based Min-Sum Algorithm to Lower the Error Floors of Quantized LDPC Decoders

   https://doi.org/10.1109/TCOMM.2020.2969902  

   Hatami, Homayoon; Mitchell, David G.; Costello, Daniel J.; Fuja, Thomas E. (January 2020, IEEE Transactions on Communications)

   For decoding low-density parity-check (LDPC) codes, the attenuated min-sum algorithm (AMSA) and the offset min-sum algorithm (OMSA) can outperform the conventional min-sum algorithm (MSA) at low signal-to-noise-ratios (SNRs), i.e., in the “waterfall region” of the bit error rate curve. This paper demonstrates that, for quantized decoders, MSA actually outperforms AMSA and OMSA in the “error floor” region, and that all three algorithms suffer from a relatively high error floor. This motivates the introduction of a modified MSA that is designed to outperform MSA, AMSA, and OMSA across all SNRs. The new algorithm is based on the assumption that trapping sets are the major cause of the error floor for quantized LDPC decoders. A performance estimation tool based on trapping sets is used to verify the effectiveness of the new algorithm and also to guide parameter selection. We also show that the implementation complexity of the new algorithm is only slightly higher than that of AMSA or OMSA. Finally, the simulated performance of the new algorithm, using several classes of LDPC codes (including spatially coupled LDPC codes), is shown to outperform MSA, AMSA, and OMSA across all SNRs. 

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5. Reed-Muller Subcodes: Machine Learning-Aided Design of Efficient Soft Recursive Decoding

   https://doi.org/10.1109/ISIT45174.2021.9517885  

   Jamali, Mohammad Vahid; Liu, Xiyang; Makkuva, Ashok Vardhan; Mahdavifar, Hessam; Oh, Sewoong; Viswanath, Pramod (July 2021, International Symposium on Information Theory and its Applications)

   Reed-Muller (RM) codes are conjectured to achieve the capacity of any binary-input memoryless symmetric (BMS) channel, and are observed to have a comparable performance to that of random codes in terms of scaling laws. On the negative side, RM codes lack efficient decoders with performance close to that of a maximum likelihood decoder for general parameters. Also, they only admit certain discrete sets of rates. In this paper, we focus on subcodes of RM codes with flexible rates that can take any code dimension from 1 to $$n$$. where $$n$$ is the blocklength. We first extend the recursive projection-aggregation (RPA) algorithm proposed recently by Ye and Abbe for decoding RM codes. To lower the complexity of our decoding algorithm, referred to as subRPA, we investigate different ways for pruning the projections. We then derive the soft-decision based version of our algorithm, called soft-subRPA, that is shown to improve upon the performance of subRPA. Furthermore, it enables training a machine learning (ML) model to search for good sets of projections that minimize the decoding error rate. Training our ML model enables achieving very close to the performance of full-projection decoding with a significantly reduced number of projections. For instance, our simulation results on a (64,14) RM subcode show almost identical performance for full-projection decoding and pruned-projection decoding with 15 projections picked via training our ML model. This is equivalent to lowering the complexity by a factor of more than 4 without sacrificing the decoding performance. 

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