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1. A Modified Min-Sum Algorithm for Quantized LDPC Decoders

   https://doi.org/10.1109/isit.2019.8849308  

   Hatami, Homayoon; Mitchell, David G.; Costello, Daniel J.; Fuja, Thomas (July 2019, 2019 IEEE International Symposium on Information Theory (ISIT))

   It is well known that 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). In this paper, we demonstrate that, for quantized LDPC decoders, although the MSA achieves better high SNR performance than the AMSA and OMSA, each of the MSA, AMSA, and OMSA all suffer from a relatively high error floor. Therefore, we propose a novel modification of the MSA for decoding quantized LDPC codes with the aim of lowering the error floor. Compared to the quantized MSA, the proposed modification is also helpful at low SNRs, where it matches the waterfall performance of the quantized AMSA and OMSA. The new algorithm is designed based on the assumption that trapping/absorbing sets (or other problematic graphical objects) are the major cause of the error floor for quantized LDPC decoders, and it aims to reduce the probability that these problematic objects lead to decoding errors. 

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2. 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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3. Quantum LDPC Codes with Enhanced Error-Floor Performance under Min-Sum Decoding

   https://doi.org/10.1109/ISTC65386.2025.11154487  

   Pacenti, Michele; Chytas, Dimitris; Vasić, Bane (August 2025, IEEE)

   Quantum low-density parity-check codes are a promising approach to fault-tolerant quantum computation, offering potential advantages in rate and decoding efficiency. Quantum Margulis codes are a new class of QLDPC codes derived from Margulis’ classical LDPC construction via the two-block group algebra framework. We show that quantum Margulis codes, unlike bivariate bicycle codes, which require ordered statistics decoding for effective error correction, can be efficiently decoded using a standard min-sum decoder with linear complexity, when decoded under depolarizing noise. This is attributed to their Tanner graph structure, which does not exhibit group symmetry, thereby mitigating the well-known problem of error degeneracy in QLDPC decoding. To further enhance performance, we propose an algorithm for constructing 2BGA codes with controlled girth, ensuring a minimum girth of 6 or 8, and use it to generate several quantum Margulis codes of length 240 and 642. We validate our approach through numerical simulations, demonstrating that quantum Margulis codes behave significantly better than BB codes in the error floor region, under min-sum decoding. 

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4. Trapping Sets of Quantum LDPC Codes

   https://doi.org/10.22331/q-2021-10-14-562  

   Raveendran, Nithin; Vasić, Bane (October 2021, Quantum)

   Iterative decoders for finite length quantum low-density parity-check (QLDPC) codes are attractive because their hardware complexity scales only linearly with the number of physical qubits. However, they are impacted by short cycles, detrimental graphical configurations known as trapping sets (TSs) present in a code graph as well as symmetric degeneracy of errors. These factors significantly degrade the decoder decoding probability performance and cause so-called error floor. In this paper, we establish a systematic methodology by which one can identify and classify quantum trapping sets (QTSs) according to their topological structure and decoder used. The conventional definition of a TS from classical error correction is generalized to address the syndrome decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be used to design better QLDPC codes and decoders. Frame error rate improvements of two orders of magnitude in the error floor regime are demonstrated for some practical finite-length QLDPC codes without requiring any post-processing. 

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5. Information Bottleneck Decoding of Rate-Compatible 5G-LDPC Codes

   https://doi.org/10.1109/ICC40277.2020.9149304  

   Stark, Maximilian; Bauch, Gerhard; Wang, Linfang; Wesel, Richard D. (June 2020, ICC 2020 - 2020 IEEE International Conference on Communications (ICC))

   The new 5G communications standard increases data rates and supports low-latency communication that places constraints on the computational complexity of channel decoders. 5G low-density parity-check (LDPC) codes have the so-called protograph-based raptor-like (PBRL) structure which offers inherent rate-compatibility and excellent performance. Practical LDPC decoder implementations use message-passing decoding with finite precision, which becomes coarse as complexity is more severely constrained. Performance degrades as the precision becomes more coarse. Recently, the information bottleneck (IB) method was used to design mutual-information-maximizing lookup tables that replace conventional finite-precision node computations. The IB approach exchanges messages represented by integers with very small bit width. This paper extends the IB principle to the flexible class of PBRL LDPC codes as standardized in 5G. The extensions include puncturing and rate-compatible IB decoder design. As an example of the new approach, a 4-bit information bottleneck decoder is evaluated for PBRL LDPC codes over a typical range of rates. Frame error rate simulations show that the proposed scheme outperforms offset min-sum decoding algorithms and operates very close to double-precision sum-product belief propagation decoding. 

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