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arXiv:2510.01019 · International Conference on Communications

Layered Normalized Min-Sum Decoding with Bit Flipping for FDPC Codes

Niloufar Hosseinzadeh, Mohsen Moradi, Hessam Mahdavifar

Published May 24, 2026• 0 citations
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Abstract

Fair-density parity-check (FDPC) codes have been recently introduced demonstrating improved performance compared to low-density parity-check (LDPC) codes standardized in 5G systems particularly in high-rate regimes. In this paper, we introduce a layered normalized min-sum (LNMS) message-passing decoding algorithm for the FDPC codes. We also introduce a syndrome-guided bit flipping (SGBF) method to enhance the error-correction performance of our proposed decoder. The LNMS decoder leverages conflict graph coloring for efficient layered scheduling, enabling faster convergence by grouping non-conflicting check nodes and updating variable nodes immediately after each layer. In the event of decoding failure, the SGBF method is activated, utilizing a novel reliability metric that combines log-likelihood ratio (LLR) magnitudes and syndrome-derived error counts to identify the least reliable bits. A set of candidate sequences is then generated by performing single-bit flips at these positions, with each candidate re-decoded via LNMS. The optimal candidate is selected based on the minimum syndrome weight. Extensive simulation results demonstrate the superiority of the proposed decoder. Numerical simulations on FDPC$(256,192)$ code with a bit-flipping set size of $T = 128$ and a maximum of $5$ iterations demonstrate that the proposed decoder achieves approximately a $0.5\,\mathrm{dB}$ coding gain over standalone LNMS decoding at a frame error rate (FER) of $10^{-3}$, while providing coding gains of $0.75-1.5\,\mathrm{dB}$ over other state-of-the-art codes including polar codes and 5G-LDPC codes at the same length and rate and also under belief propagation decoding.

DOI
10.1109/icc59461.2026.11587382

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Fair-density parity-check (FDPC) codes have been recently introduced demonstrating improved performance compared to low-density parity-check (LDPC) codes standardized in 5G systems particularly in high-rate regimes. In this paper, we introduce a layered normalized min-sum (LNMS) message-passing decoding algorithm for the FDPC codes. We also introduce a syndrome-guided bit flipping (SGBF) method to enhance the error-correction performance of our proposed decoder. The LNMS decoder leverages conflict graph coloring for efficient layered scheduling, enabling faster convergence by grouping non-conflicting check nodes and updating variable nodes immediately after each layer. In the event of decoding failure, the SGBF method is activated, utilizing a novel reliability metric that combines log-likelihood ratio (LLR) magnitudes and syndrome-derived error counts to identify the least reliable bits. A set of candidate sequences is then generated by performing single-bit flips at these positions, with each candidate re-decoded via LNMS. The optimal candidate is selected based on the minimum syndrome weight. Extensive simulation results demonstrate the superiority of the proposed decoder. Numerical simulations on FDPC(256,192)(256,192)(256,192) code with a bit-flipping set size of T=128T = 128T=128 and a maximum of 555 iterations demonstrate that the proposed decoder achieves approximately a 0.5 dB0.5\,\mathrm{dB}0.5dB coding gain over standalone LNMS decoding at a frame error rate (FER) of 10−310^{-3}10−3, while providing coding gains of 0.75−1.5 dB0.75-1.5\,\mathrm{dB}0.75−1.5dB over other state-of-the-art codes including polar codes and 5G-LDPC codes at the same length and rate and also under belief propagation decoding.
Entering the 5G era, the hardware design of low-density parity-check (LDPC) codes under the new standard has increasingly higher requirements on throughput. According to the geometric characteristics of the check matrices of the quasi-cyclicQC) LDCP codes under the CCSDS standard, this paper designs a layered dynamic normalized minimum sum algorithm(LDNMSA) under the pre-termination decoding based on de-layering scheme. The check matrix structure of this code is easy for hardware design and the storage resource consumption is low. The approximate replacement of the minimum sum algorithm(MSA) reduces the computational complexity of the check node update process. The determination of correction factors and inter-layer deletion thresholds through computer simulation improves the decoding performance and the speed of a single iteration, and adopt the layered scheduling scheme with optimized update order reduces the number of iterations required for decoding. Experimental results show that when the bit error rate(BER) is 10-5, the designed algorithm has a gain of approximately 0.5 dB compared to the MSA. The speed and calculation amounts to a single iteration are much lower than the log-likelihood-ratio(LLR) belief propagation(BP) algorithm and the performance is only less than 0.1dB.
Normalized Min-Sum Algorithms (NMSA) are extensively employed in trading LDPC (Low Density Parity Check) decoders due to their low complexity and reasonable performance close to the performance of sum product algorithm in terms of Error Rate. Our work is focused on the enhancement of error rate performances of NMS algorithm for the decoding of LDPC decoding. In this context we have proposed a Layered Normalized Min-Sum algorithm (LNMS), which employs an adaptive normalization factor to ameliorate the reliability of the information transmitted during the decoding process. To perform this analysis study LDPC coded communication system has been examined over Weibull fading channels and was analyzed for different decoding techniques by the help of comparative computer simulations. Simulation results indicate that the proposed decoding algorithm can achieve better Bit Error Rate (BER) performances closer to SP algorithm with the preservation of the principal characteristics of the MS algorithm.
In order to improve the bit-flipping (BF) decoding algorithms, a weighted hard-decision based BF decoding algorithm for low-density parity-check (LDPC) codes is presented, in which the weights of variable-nodes are employed. A new metric is defined for the weight of candidate bits not only in unsatisfied parity check equations but also in satisfied ones. The connection between min-sum decoding algorithm and the proposed algorithm is presented. Simulation results show that the new decoding algorithm can provide better FER performance and faster convergence speed than the original candidate bit based bit-flipping (CBBF) algorithm.
Recently introduced Fair-Density Parity-Check (FDPC) codes, targeting high-rate applications, offer superior error-correction performance (ECP) compared to 5G Low-Density Parity-Check (LDPC) codes, given the same number of message-passing decoding iterations. In this paper, we present a novel construction method for FDPC codes, introduce a generalization of these codes, and propose a low-complexity encoding algorithm. Numerical results demonstrate the fast convergence of the message-passing decoder for FDPC codes.
In this paper, a modified normalized Min-Sum algorithm (MNMSA) is proposed for decoding irregular Low Density Parity Check Codes (LDPC) with non-uniform degree distribution. The proposed algorithm provides an efficient decoding scheme to enhance the error performance of an irregular LDPC code without increasing the hardware complexity. An efficient 6- bit quantization scheme is utilized in the proposed algorithm with optimally combined normalization and down scaling factors to resolve the magnitude overestimation issue which occurs during iterative message passing process between the nodes. The simulation results demonstrate that the proposed algorithm achieves good decoding performance in terms of Frame Error Rate (FER) and Bit Error Rate (BER) without the requirement of complex arithmetic calculations. Compared to other min- sum based decoding algorithms the decoding process of the proposed MNMSA requires only fewer decoding iterations over wide range of signal- to- noise ratio (SNR) to achieve comparable decoding performance to that of Belief Propagation (BP) algorithm without increasing the hardware complexity.
We introduce fair-density parity-check (FDPC) codes targeting high-rate applications. In particular, we start with a base parity-check matrixHbH_{b}Hb​of dimension2n×n2\sqrt{n}\times n2n​×n, wherennnis the code block length, and the number of ones in each row and column ofHbH_{b}Hb​is equal ton\sqrt{n}n​and 2, respectively. We propose a deterministic combinatorial method for picking the base matrixHbH_{b}Hb​, assumingn=4t2n=4t^{2}n=4t2for some integert⩾2t\geqslant 2t⩾2. We then extend this by obtaining permuted versions ofHbH_{b}Hb​(e.g., via random permutations of its columns) and stacking them on top of each other leading to codes of dimensionk⩾n−2sn+sk\geqslant n-2s\sqrt{n}+sk⩾n−2sn​+s, for somes⩾2s\geqslant 2s⩾2, referred to as order-s FDPC codes. We propose methods to explicitly characterize and bound the weight distribution of the new codes and utilize them to derive union-type approximate upper bounds on their error probability under Maximum Likelihood (ML) decoding. For the binary erasure channel (BEC), we demonstrate that the approximate ML bound of FDPC codes closely follows the random coding upper bound (RCU) for a wide range of channel parameters. Also, remarkably, FDPC codes, under the low-complexity min-sum decoder, improve upon 5G-LDPC codes for transmission over the binary-input additive white Gaussian noise (B-AWGN) channel by almost 0.5dB (forn=1024n=1024n=1024, and rate = 0.878). Furthermore, we propose a new decoder as a combination of weighted min-sum message-passing (MP) decoding algorithm together with a new progressive list (PL) decoding component, referred to as the MP-PL decoder, to further boost the performance of FDPC codes. This paper opens new avenues for a fresh investigation of new code constructions and decoding algorithms in high-rate regimes suitable for ultra-high throughput (high-frequency/optical) applications.