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HARS: LDPC Bit-Flipping Decoding With Initial-Syndrome-Conditioned Parameter Mapping and Local-Reliability Weighting

Tung Hsu Wu, Huang-Chang Lee

arXiv:2609.36727v1cs.ITmath.IT

TL;DR

Bit-flipping decoders must select reliable nodes without the soft-message storage used by belief propagation. HARS combines bounded check-level reliability with parameters selected from each frame’s initial syndrome, and reports lower BER and mean iteration counts than calibrated SM-NGDBF on PEGReg and WiMAX. Its FPGA implementation reaches 361.9 Mb/s at 4.00 dB, including initialization and output.

  • Problem

    Bit-flipping decoding needs enough reliability information for accurate node selection while keeping updates suitable for parallel hardware.

    Source: §I · p. 1

  • Method

    HARS weights parity checks using bounded local reliability and uses the initial syndrome to select four frame parameters before synchronous multi-bit updates.

    Source: §I · p. 1

  • Results

    HARS reduces BER and mean iteration count relative to calibrated SM-NGDBF on PEGReg and WiMAX; PEGReg BER at 3.25 dB is about one-sixth of the SM-NGDBF value.

    Source: B. PEGReg Results · p. 5, §VI · p. 13

  • Takeaways & Limitations

    The XC7A200T implementation reaches 361.9 Mb/s at 4.00 dB and 60 MHz, including initialization and output, using four contexts on a shared 132-lane pipeline.

    Source: §VI · p. 13

Abstract

from arXiv · show

HARS (Hybrid Adaptive Reliability-Aware and Syndrome-Aware) is a bit-flipping decoder that combines local channel reliability with parameter selection from the initial syndrome. A bounded check reliability modifies the parity-check weight, while the initial syndrome weight selects the base weight, reliability coefficient, threshold decay factor, and perturbation amplitude for each frame. Computing these quantities at initialization supports synchronous multi-bit updates with a simple iterative datapath. For rate-one-half PEGReg and WiMAX codes over the binary-input additive white Gaussian noise channel, HARS reduces both bit-error rate (BER) and mean iteration count relative to a calibrated SM-NGDBF baseline. The PEGReg BER at 3.25 dB is approximately one-sixth of the comparison value; the WiMAX BER at 2.75 dB is reduced by 59%. A seven-fractional-bit FPGA implementation shares 132 variable-node lanes among four frame contexts. Precomputed check weights, encoded threshold states, and parallel perturbation generation support one group per clock during sustained decoding. On an XC7A200T at 60 MHz, for preloaded 128-frame batches with a ready receiver, measured coded throughputs are 49.7 Mb/s at 2.50 dB and 361.9 Mb/s at 4.00 dB, including initialization and output transfers.

I. Introduction

Bit-flipping decoding avoids soft-message exchange, but must extract enough reliability for accurate node selection while remaining suitable for parallel hardware. HARS combines check-level reliability with frame-level parameter selection from the initial syndrome, and separates initialization from iteration to simplify its hardware datapath.

I. Introduction

  • HARS weights each check using bounded local reliability, while the initial syndrome selects four frame parameters before decoding begins.The frame parameters remain fixed during iterations; local reliability differentiates checks within the frame.Source: §I · p. 1
  • Prior noisy bit-flipping methods use inversion histories, syndrome information, or modified perturbations, whereas HARS uses the initial syndrome and check reliability at initialization.After initialization, decisions, syndromes, and node thresholds evolve through synchronous updates.Source: §I · p. 1
  • Precomputed check weights and encoded threshold states remove repeated multiplication from the iterative datapath, which shares 132 variable-node lanes among four frame contexts.Double-buffered decisions and syndromes preserve synchronous updates across the shared pipeline.Source: §I · p. 1

A. Channel, Decisions, and Syndromes

  • The paper evaluates HARS on rate-one-half PEGReg and WiMAX codes over the binary-input additive white Gaussian noise channel.The experiments transmit the all-zero codeword and count bit and frame errors against it, separately from the parity-based stopping test.Source: A. Channel, Decisions, and Syndromes · p. 2

B. From GDBF to Smoothed Multi-Bit NGDBF

GDBF combines channel observations with parity constraints in an inversion metric, while noisy multi-bit variants perturb node selection and adapt local thresholds. HARS extends this approach with bounded check-level reliability weighting.

B. From GDBF to Smoothed Multi-Bit NGDBF

  • GDBF combines channel agreement and parity-check information in an objective whose channel terms measure agreement and whose check terms indicate constraint satisfaction.Flipping a bit reverses its channel term and all adjacent check terms, defining its inversion metric.Source: B. From GDBF to Smoothed Multi-Bit NGDBF · p. 2
  • NGDBF adds fresh perturbations so nodes near a flip boundary can be selected differently, and multi-bit variants synchronously flip all nodes below their evolving local thresholds.SM-NGDBF uses a vote over terminal decisions when the iteration limit is reached; a zero syndrome instead stops decoding immediately.Source: B. From GDBF to Smoothed Multi-Bit NGDBF · p. 2

A. Local-Reliability Weighting

  • HARS weights each parity-check contribution using bounded reliability derived from the weakest adjacent channel observation, while preserving the base contribution when that magnitude is small.The correction is clipped to a fixed range for hardware representation.Source: A. Local-Reliability Weighting · p. 2
  • Reliable satisfied checks favor retaining a decision, while reliable unsatisfied checks favor flipping, so equal unsatisfied-check counts can yield different node selections.The effective check weight lies in [w, w + α] for α ≥ 0.Source: A. Local-Reliability Weighting · p. 2

B. Initial-Syndrome-Conditioned Parameter Mapping

HARS uses each frame’s initial syndrome weight to select its base check weight, local-reliability coefficient, threshold decay factor, and perturbation amplitude. The mappings are calibrated separately for each code and clipped to their specified limits.

B. Initial-Syndrome-Conditioned Parameter Mapping

  • Each frame’s initial syndrome weight selects clipped linear mappings for the base weight, reliability coefficient, threshold decay factor, and perturbation amplitude.All four mapped parameters are evaluated at initialization and shared by every node and update in that frame.Source: B. Initial-Syndrome-Conditioned… · pp. 2–3
  • For WiMAX, increasing S0 reduces w, α, and η, while increasing λ slows the movement of negative thresholds toward zero.The mappings jointly adjust the inversion metric and threshold evolution; Fig. 1 shows the mapped values and clipping limits.Source: B. Initial-Syndrome-Conditioned… · p. 3, Figure 1 · p. 3
  • The mean and median S0 decrease as Eb/N0 increases, but received frames span a range of initial syndrome weights at each operating point.Frame-specific mapping uses this variation, while ρ_i retains check-level channel information.Source: B. Initial-Syndrome-Conditioned… · p. 3

C. Parameter Formation and Code-Specific Settings

  • The mapping coefficients are calibrated separately for each code; α increases with S0 for PEGReg and decreases with S0 for WiMAX.PEGReg calibration uses successive coordinate-wise grid searches, then fits clipped linear mappings from better-performing parameter candidates and corresponding S0 data; coefficients stay fixed during BER evaluation.Source: B. Initial-Syndrome-Conditioned… · p. 3, C. Parameter Formation and… · p. 3
  • WiMAX uses its own calibration with the same HARS reliability measure, mapping form, and synchronous update rule as PEGReg.The WiMAX QC structure also organizes the hardware interconnect.Source: C. Parameter Formation and… · p. 3

D. Inversion Metric and Synchronous Flipping

HARS forms an inversion metric from channel information and weighted parity-check contributions, then evaluates all nodes from a common pre-update state before committing flips synchronously. Thresholds evolve with node history, and terminal decisions are smoothed when the iteration limit is reached.

D. Inversion Metric and Synchronous Flipping

  • HARS combines local reliability weighting and frame-level control in its inversion metric, with fresh standard-normal perturbations drawn for each node and iteration.Frame parameters follow the initial-syndrome mappings, and λ controls the threshold recurrence.Source: D. Inversion Metric and Synchronous… · p. 3
  • The base and reliability check terms use the same syndrome sign, so local reliability changes parity-information strength without changing its direction.All node comparisons use the pre-update decisions and thresholds; the complete flip set is committed synchronously before syndrome recomputation.Source: D. Inversion Metric and Synchronous… · p. 4

E. Threshold Evolution and Terminal Smoothing

  • For a retained node, 0 < λ < 1 moves its negative threshold toward zero, while a flipped node returns to θ_reset.This gradually relaxes the flip condition for retained nodes.Source: E. Threshold Evolution and Terminal… · p. 4
  • A zero syndrome returns the current decisions immediately; if T_max is reached with a nonzero syndrome, the decoder votes over the final 64 committed decisions.Ties retain the decision from update T_max, and the final syndrome is recomputed after voting.Source: E. Threshold Evolution and Terminal… · p. 4

F. State and Computational Structure

HARS separates fixed frame and check quantities from evolving decoder state, with per-iteration work linear in edge and node counts. Evaluation spans two rate-one-half codes and specifies how BER and mean iterations are measured.

F. State and Computational Structure

  • Four frame parameters and M check reliabilities remain fixed after initialization; evolving state holds N decisions, thresholds, syndromes, and per-bit smoothing counters.The four frame parameters and check reliabilities are fixed, while the decisions, thresholds, syndromes, and smoothing counters evolve.Source: F. State and Computational Structure · p. 4
  • At fixed word length, initialization and I iterations require O((I+1)(E_H+N+M)) operations per frame, including initialization.Each update scans neighborhoods for weighted checks and syndrome recomputation, alongside N metric, threshold, and decision operations.Source: F. State and Computational Structure · p. 4

A. Experimental Setup

  • Tests use rate-one-half PEGReg (1008, 504) and WiMAX (1056, 528), with floating-point Eb/N0 sweeps from 2.0 to 4.0 dB in 0.25-dB steps.WiMAX uses a 12 × 24 base matrix with circulant size Z = 44.Source: A. Experimental Setup · p. 4
  • Methods use common channel-frame sequences at each operating point, with channel noise generated separately from decoder perturbations; BER is errors divided by decoded bits.Mean iterations include every decoded frame, assigning zero updates to an initially zero syndrome and Tmax to a frame reaching the limit.Source: A. Experimental Setup · p. 4
  • The SM-NGDBF comparison follows perturbation and smoothing, but resets its threshold after a flip; HARS and the comparison variant share Tmax = 300, Ymax = 2.5, and a 64-decision smoothing window.The original rule retains the threshold on a flip.Source: A. Experimental Setup · pp. 4–5
  • PEGReg includes S-A-NGDBF and NMS references, while WiMAX additionally includes S-NGDBF; both HARS and the comparison variant use a 300-iteration limit.The WiMAX S-NGDBF settings include check weight 0.732 and noise scale 0.678.Source: A. Experimental Setup · p. 5

B. PEGReg Results

Across PEGReg and WiMAX, HARS improves BER relative to SM-NGDBF, and it also requires fewer mean iterations. The PEGReg ablation attributes the larger BER improvement to local reliability, with an additional reduction from frame adaptation.

B. PEGReg Results; C. WiMAX Results

  • PEGReg BER at 3.25 dB is about one-sixth of SM-NGDBF’s value, while WiMAX BER at 2.75 dB is reduced by approximately 59% versus SM-NGDBF.HARS has the lowest BER among the three bit-flipping methods across the PEGReg sweep and outperforms SM-NGDBF and S-NGDBF across the tested WiMAX range.Source: B. PEGReg Results · p. 5, Figure 3 · p. 5, C. WiMAX Results · p. 5
  • At 3 dB on PEGReg, HARS reduces mean iterations by approximately 41% versus SM-NGDBF; on WiMAX it uses fewer iterations at all nine points, with a 13% reduction at 2.75 dB.The two WiMAX multi-bit curves converge as SNR increases; S-NGDBF uses single-bit updates.Source: B. PEGReg Results · p. 5, C. WiMAX Results · p. 5, Figure 5 · p. 6

D. Ablation of Local Reliability and Frame Adaptation

  • Local reliability accounts for the main BER reduction; frame-dependent selection further reduces BER by approximately 22% at 3.25 dB, with full HARS below the fixed-parameter curve from 3.25 to 4 dB.At 3 dB, the 5th–95th percentiles of S0 span 139–185 around the mean used for the fixed setting; frame adaptation adjusts parameters across this range before local threshold comparisons determine the flip set.Source: D. Ablation of Local Reliability and… · p. 6
  • In a separate 3 dB fixed-length comparison, frame adaptation reduces empirical BER by 4.6% relative to local reliability with fixed parameters.The configurations were unretuned and used 100,000 common channel frames and base perturbation sequences.Source: D. Ablation of Local Reliability and… · p. 6

E. Fixed-Point Precision Selection

F7 is selected because it improves PEGReg/WiMAX precision results over lower-precision configurations. The architecture moves frame-specific preparation into initialization and shares 132 VN lanes across four contexts.

E. Fixed-Point Precision Selection

  • At 4.00 dB, F7 achieves approximately 4.60 × 10−6 BER, about one-nineteenth of F3 and 19% below F6.This comparison motivates selecting F7 for the architecture.Source: E. Fixed-Point Precision Selection · p. 6

V. Partially Parallel HARS Architecture

  • Four frame contexts share 132 processing lanes, while precomputed check weights and encoded threshold states remove multiplication from the iterative datapath.Double-buffered decisions and syndromes preserve synchronous updates across the shared pipeline.Source: §V · p. 6

A. Architecture Overview and Streaming Initialization

1) Streaming initialization:

  • Initialization accepts 22 ten-bit channel samples per transfer and uses six 22-lane barrel shifters to align samples with check accumulators.A frame comprises 48 successful sample transfers; admission and stalls follow the interface handshake.Source: 1) Streaming initialization · p. 7
  • After the final sample transfer, the initial syndrome population selects four frame parameters, and 22 DSP blocks compute 528 check-node weights for storage.Signed weights are stored for four contexts, so no weight multiplication is needed during decoding.Source: 1) Streaming initialization · p. 8

2) QC mapping:

The selected WiMAX QC matrix fixes the VN-to-check contribution slots, while threshold lookup, parallel perturbation generation, and buffered state updates support synchronous decoding.

2) QC mapping:

  • Paired block rows have disjoint supports, so each VN connects to at most one check in each pair and fits six contribution slots.Unused slots receive zero, and the QC connections are fixed at elaboration.Source: 2) QC mapping · p. 8, Figure 9 · p. 9

B. Variable-Node Datapath and Threshold Lookup

  • Threshold lookup encodes each frame coefficient’s rounded decay trajectory as reachable states, with a flip selecting reset and a retained decision selecting the successor.The circuit stores a six-bit index per node, and at most 56 states are needed per coefficient.Source: B. Variable-Node Datapath and… · p. 9

C. Parallel Perturbations and Synchronous State Updates

  • Four LFSRs provide bits whose population count forms a five-level perturbation, then sign and shift selection scales it by the frame-dependent amplitude.Only accepted decoding requests advance generator states, preserving each frame’s sequence across scheduling positions.Source: C. Parallel Perturbations and… · pp. 9–10
  • Double-buffered decisions and syndromes let every group in an iteration read the same committed state while updated decisions and syndrome contributions are accumulated for writeback.After the final contribution, buffer roles are exchanged and the completed syndrome is tested.Source: C. Parallel Perturbations and… · pp. 9–10

D. Frame-Interleaved Scheduling

Frame interleaving hides iteration-boundary writeback dependencies by scheduling other ready contexts while a frame’s updates complete.

D. Frame-Interleaved Scheduling

  • The scheduler accepts one group per clock during sustained decoding, then issues work from another ready context while prior iteration updates finish writing back.An eight-group iteration’s final updates remain in the five-clock pipeline after the groups are issued.Source: D. Frame-Interleaved Scheduling · p. 11
  • Initially zero syndromes bypass iteration, zero syndromes after an update go to output, and frames reaching iteration 300 with nonzero syndrome use smoothing.Output, smoothing, and decoding share the state-read path, and their arbitration is included in elapsed batch time.Source: D. Frame-Interleaved Scheduling · p. 11

E. Fixed-Point Arithmetic

Fixed-point arithmetic represents the selected HARS mappings and datapath with quantized codes, rounded threshold updates, and a bounded reachable threshold range.

E. Fixed-Point Arithmetic

  • F7 uses seven fractional bits for channel samples and the VN metric, while the four parameter mappings retain ten fractional bits.Parameter ROMs addressed by S0 implement the mappings during frame initialization.Source: E. Fixed-Point Arithmetic · p. 11
  • For WiMAX, the complete F3 amplitude mapping produces integer code 5, while F7 retains codes 77–88 so frame-to-frame η differences affect applied perturbations.Threshold recursion also depends on the fixed-point step size; a decay step can round back to the same value.Source: E. Fixed-Point Arithmetic · p. 11
  • The threshold code is stored in eight bits because its reachable values lie in [−128, −3], and the signed 17-bit sum can be compared directly with the sign-extended threshold.The reachable threshold range is inside the 16-bit saturation interval.Source: E. Fixed-Point Arithmetic · p. 11

F. FPGA Results and Comparison

FPGA measurements show higher coded throughput at 4.00 dB than at 2.50 dB, with both values including initialization and output. HARS maps its reliability-weighted update to a shared, synchronous 132-lane pipeline, and its conclusion reports reduced BER and mean iteration count relative to calibrated SM-NGDBF.

F. FPGA Results and Comparison

  • Across seven SNR points, RTL matched the fixed-point reference on 128 quantized inputs per point in decoded words and recorded status fields.Directed tests also exercised output stalls, initially zero syndromes, and context reuse.Source: F. FPGA Results and Comparison · p. 11
  • 361.9 Mb/s at 4.00 dB versus 49.7 Mb/s at 2.50 dB; both measured 128-frame batch throughputs include initialization and output.The comparison is shown in the 60 MHz coded-throughput figure and the 128-frame batch performance table.Source: F. FPGA Results and Comparison · p. 12, Figure 12 · p. 12, Table VI · p. 12
  • The throughput comparison with AWGN MNGDBF uses different timing bases: MNGDBF reports 100 iterations, while HARS uses measured batches including initialization and output.The table specifies each implementation's code, precision, and timing basis.Source: F. FPGA Results and Comparison · p. 12, Table VII · p. 13

VI. Conclusion

  • HARS reduces BER and mean iteration count relative to calibrated SM-NGDBF on the evaluated PEGReg and WiMAX configurations.In the PEGReg ablation without retuning, local weighting provides the larger BER improvement, with a further reduction from frame adaptation.Source: §VI · p. 13
  • Four frame contexts share 132 lanes while preserving synchronous state updates; precomputed weights and encoded threshold trajectories remove repeated multiplication from the VN update.The XC7A200T implementation reaches 361.9 Mb/s at 4.00 dB and 60 MHz, including initialization and output.Source: §VI · p. 13
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