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Enhanced Precision Through Multiple Reads for LDPC Decoding in Flash Memories

Jiadong Wang, Kasra Vakilinia, Tsung-Yi Chen, Thomas Courtade, Guiqiang Dong, Tong Zhang, Hari Shankar, Richard Wesel

arXiv:1309.0566v4cs.ITmath.IT

TL;DR

As Flash levels increase, channel sensitivity motivates stronger error correction, and the paper examines how multiple reads can provide LDPC decoders with enhanced precision. It optimizes read voltages using mutual information and reports performance benefits alongside code- and precision-dependent qualifications.

  • Problem

    Increasing Flash levels heightens sensitivity to SNR variation, making stronger error-correction codes necessary.

    Source: §I · p. 1

  • Method

    The paper optimizes word-line voltages for enhanced-precision reads by maximizing mutual information of the quantized read channel, with a constant-ratio constraint to simplify optimization.

    Source: §VI · p. 11

  • Results

    A few additional reads provide most of the performance benefit, and simulations suggest MI-maximizing quantization minimizes FER for well-designed LDPC codes but may not for codes with high error floors.

    Source: §VI · p. 11

  • Takeaways & Limitations

    The best LDPC code can differ across precision levels, so code design must account for decoding at varying levels of precision.

    Source: B. Quantization-based Design Trade-off · p. 6

    The simulations use floating-point decoder messages; limited-precision message representations alongside limited-precision channel information remain for further investigation.

    Source: §VI · p. 11

Abstract

from arXiv · show

Multiple reads of the same Flash memory cell with distinct word-line voltages provide enhanced precision for LDPC decoding. In this paper, the word-line voltages are optimized by maximizing the mutual information (MI) of the quantized channel. The enhanced precision from a few additional reads allows FER performance to approach that of full-precision soft information and enables an LDPC code to significantly outperform a BCH code. A constant-ratio constraint provides a significant simplification in the optimization with no noticeable loss in performance. For a well-designed LDPC code, the quantization that maximizes the mutual information also minimizes the frame error rate in our simulations. However, for an example LDPC code with a high error floor caused by small absorbing sets, the MMI quantization does not provide the lowest frame error rate. The best quantization in this case introduces more erasures than would be optimal for the channel MI in order to mitigate the absorbing sets of the poorly designed code. The paper also identifies a trade-off in LDPC code design when decoding is performed with multiple precision levels; the best code at one level of precision will typically not be the best code at a different level of precision.

I. INTRODUCTION

As Flash storage density grows, stronger error correction is needed, while the paper investigates how multiple reads can supply LDPC decoders with enhanced precision. It optimizes read voltages using mutual information and examines the benefits and design trade-offs across precision levels and Flash models.

I. INTRODUCTION

  • Increasing Flash levels heightens sensitivity to SNR variation, motivating stronger error-correction codes and adaptive coding or modulation.Smaller feature sizes can also make inter-cell interference more likely.Source: §I · p. 1

A. Related Work

  • The paper focuses on enhanced precision for standard LDPC codes, while noting that soft information may also benefit schemes targeting dominant cell-level errors.It distinguishes this focus from approaches that address wear-out or use rank modulation,.Source: A. Related Work · pp. 1–2

B. Overview and Contributions

  • Multiple reads with distinct word-line voltages provide LDPC decoders enhanced precision, addressing Flash systems' historically hard-decision-only input.The paper studies voltage selection, the number of additional reads, and how precision levels affect code design.Source: B. Overview and Contributions · p. 2
  • A few additional reads provide most of the enhanced-precision benefit in both mutual-information analysis and an LDPC simulation example.The paper optimizes word-line voltages to maximize mutual information for the equivalent read channel.Source: B. Overview and Contributions · p. 2
  • The best LDPC code at one precision level is not necessarily best by density-evolution threshold or empirical performance at another.Precision may vary because additional page reads are likely to be used when a page cannot be decoded without them.Source: B. Overview and Contributions · p. 2
  • For four-level MLC, the paper uses a realistic channel model and finds that MI-maximizing quantization can minimize FER for a well-designed code, but not for a poorly designed one.For the latter, larger erasure regions than those optimal for MI are needed to mitigate small absorbing sets.Source: B. Overview and Contributions · p. 2

II. THE READ CHANNEL OF NAND FLASH MEMORY

The NAND read channel measures cell charge indirectly: a word-line voltage and sense-amp comparator yield a one-bit decision, while threshold-voltage variation introduces read-channel noise. The paper begins with an i.i.d. Gaussian SLC model and also considers a more realistic MLC model.

II. THE READ CHANNEL OF NAND FLASH MEMORY

  • A read applies a word-line voltage and compares drain current with a threshold, yielding one bit about the cell's charge level.The threshold voltage can vary because of write overcharge, retention leakage, or charge from nearby cells; this variation is read-channel noise.Source: §II · p. 3
  • The initial SLC model assumes i.i.d. Gaussian threshold voltages, equivalent to BPSK over AWGN except that the threshold voltage is not directly observed.At most one bit of information about that voltage is obtained per cell read.Source: §II · p. 3
  • For a more realistic MLC analysis, the paper uses the model of, which accounts for inter-cell interference, program injection statistics, random telegraph noise, and retention noise.The paper contrasts this with simpler Gaussian approximations.Source: §II · p. 3

III. SOFT INFORMATION VIA MULTIPLE CELL READS

Flash sense-amp comparators traditionally provide hard decisions, but repeated reads at selected word-line voltages can yield soft information. The paper chooses those voltages by maximizing mutual information for quantized read channels, illustrating the approach with two- and three-read SLC models.

A. Obtaining Soft Information

  • Multiple reads using the same sense-amp comparator and different word-line voltages can provide additional useful information when the voltages are well chosen.The paper focuses on this approach; multiple comparators may be less informative when the transistor I-V curve is too nonlinear.Source: A. Obtaining Soft Information · p. 3
  • The approach selects word-line voltages by maximizing mutual information between the input and output of the equivalent discrete-alphabet read channel.For the quantization, this choice maximizes the amount of information provided.Source: A. Obtaining Soft Information · p. 3

B. Quantizing Flash to Maximize Mutual Information

  • For SLC threshold voltages modeled as identically distributed Gaussian, Fig. 2 shows two-read word-line voltages as dashed lines and depicts quantization regions, with the middle region comprising blue and purple regions.Source: Figure 2 · p. 4
  • Figure 3 presents equivalent discrete memoryless channel models for SLC under two-read and three-read configurations, with the three-read model using distinct word-line voltages.Source: Figure 3 · p. 4

1) Two reads per cell:

For two reads per SLC cell, symmetric thresholds q and −q divide the threshold voltage into three regions and produce a discrete memoryless channel. The threshold is chosen to maximize the channel’s mutual information.

1) Two reads per cell:

  • Two symmetric reads at q and −q quantize the threshold voltage into three regions, including an erasure region, and produce a channel with outputs 0, e, and 1.At q = 0 the erasure region vanishes, equivalent to a single read.Source: 1) Two reads per cell · p. 4
  • With equally likely inputs, the mutual information between the channel input X and output Y is computed from the resulting discrete memoryless channel probabilities.The probabilities are expressed using the Gaussian Q function.Source: 1) Two reads per cell · p. 4
  • The mutual information is quasi-concave in q, with one finite stationary point giving its maximum; a numerical bisection search finds that threshold.The derivative becomes negative at sufficiently large q and approaches zero as q tends to infinity.Source: 1) Two reads per cell · p. 4

2) Three reads per cell:

For three reads per SLC cell, symmetric thresholds q, 0, and −q create four quantization regions and a four-output discrete memoryless channel. The mutual information reaches its maximum at an interior threshold.

2) Three reads per cell:

  • Three symmetric reads at q, 0, and −q divide the threshold voltage into four regions and yield channel outputs 00, 01, 10, and 11.The input is binary, and the channel probabilities are defined using the Gaussian Q function.Source: 2) Three reads per cell · pp. 4–5
  • The mutual information has the same value at q = 0 and q = ∞, where the channel is equivalent to a single-read binary symmetric channel.Between these extremes, the derivative has one zero corresponding to the maximum mutual information.Source: 2) Three reads per cell · p. 5
  • The maximizing threshold can be found numerically by solving for the zero of the mutual-information derivative, for example with bisection.For realistic, numerically specified distributions, bisection can also find the optimum; non-identical distributions can use constant-ratio single-parameter optimization.Source: 2) Three reads per cell · p. 5

C. Performance vs. Number of Reads Per Cell

The paper evaluates how additional reads affect mutual information and LDPC frame-error performance in an SLC Gaussian channel. A few additional reads provide most of the benefit, while gains diminish as reads increase.

C. Performance vs. Number of Reads Per Cell

  • For more than three reads, MMI optimization becomes more complex because MI need not be concave or quasi-concave in multiple parameters.The authors use a coarse brute-force search followed by bisection in promising regions.Source: C. Performance vs. Number of Reads Per… · p. 5
  • Each additional read improves LDPC Code 2’s FER with diminishing gains; three reads bring its performance close to its full-soft-information limit, and the LDPC code outperforms the BCH code even with one read.Both codes have rate 0.9021, and the BCH and one-read LDPC curves use hard decoding.Source: Figure 6 · p. 5, C. Performance vs. Number of Reads Per… · p. 6
  • Two reads close about half the mutual-information gap between single-read quantization and full soft information, with diminishing gains from further reads.As reads increase, the channel bit-error probability required to reach MI 0.9021 also increases.Source: C. Performance vs. Number of Reads Per… · p. 5, Figure 5 · p. 5

IV. LDPC CODE DESCRIPTIONS

LDPC codes use sparse parity-check matrices and iterative belief-propagation decoding, where soft reliability information can improve performance. The described codes illustrate that relative code performance depends on precision and that MMI quantization may not minimize FER for a poorly designed code.

IV. LDPC CODE DESCRIPTIONS

  • LDPC codes use sparse parity-check matrices and low-complexity iterative belief-propagation decoding, which commonly relies on soft reliability information.A quantization that is too coarse can degrade performance.Source: §IV · p. 6

A. Description of LDPC Codes

  • Unlike BCH codes, LDPC codes do not readily guarantee a fixed number of correctable errors, though their average bit-error-rate performance can outperform BCH codes in Gaussian noise.The three simulated LDPC codes use matrices constructed with ACE and stopping-set checks, and decoding runs for up to 50 iterations.Source: §IV · p. 6, A. Description of LDPC Codes · p. 6
  • Codes 2 and 3 reverse their density-evolution threshold ranking between single-read and full-precision AWGN channels.Code 2 has the better single-read threshold, while Code 3 has the better full-precision threshold.Source: A. Description of LDPC Codes · p. 6
  • Code 2’s MMI quantization minimizes FER because the code avoids the small absorbing sets that make Code 1’s MMI thresholds suboptimal.For Code 1, the best quantization instead uses more erasures to mitigate the absorbing sets.Source: A. Description of LDPC Codes · p. 6
  • Codes 1 and 3 optimize full-precision AWGN density-evolution thresholds, while Code 2 modifies Code 1’s degree distribution and has a better single-read threshold than either code.Codes 1 and 3 have maximum variable-node degrees 19 and 24, respectively.Source: A. Description of LDPC Codes · p. 6

B. Quantization-based Design Trade-off

LDPC code rankings shift with decoding precision, while the MLC extension uses MI-based threshold selection and finds that six reads recover much of the gap to full soft decoding. A single-parameter constraint has insignificant MI cost, but Code 1’s error floor illustrates that code structure can alter FER outcomes.

B. Quantization-based Design Trade-off

  • Because page reads are time-intensive and may be added only when decoding requires them, a code may need to perform across several precision levels.This motivates the code-design trade-off considered in the section.Source: B. Quantization-based Design Trade-off · p. 6

AND SNR. SINGLE-READ THRESHOLDS ARE IN TERMS OF CHANNEL BIT

  • Code 2 outperforms Code 3 with one read, Code 3 outperforms Code 2 with full soft information and three reads, and the codes perform essentially alike with two reads.The density-evolution results also show the single-read/full-precision threshold trade-off.Source: AND SNR. SINGLE-READ THRESHOLDS ARE IN… · p. 7, Figure 7 · p. 7

A. MMI Quantization for MLC

  • For four-level MLC, six reads create a four-input, seven-output channel, and the three added thresholds are selected by maximizing mutual information.The thresholds are q1, q2, and q3.Source: A. MMI Quantization for MLC · p. 7, §V · p. 7
  • Constraining the three erasure regions to equal size around the hard-decoding thresholds makes optimization a single-parameter problem, with insignificant benefit from fully unconstrained optimization.The single parameter is q, and the figure compares constrained and unconstrained optimization.Source: A. MMI Quantization for MLC · pp. 7–8, Figure 10 · p. 8
  • Six reads recover more than half the performance gap between three-read hard decoding and full soft-precision decoding for four-level MLC Flash.The comparison uses unconstrained MMI quantization on the Gaussian channel model.Source: A. MMI Quantization for MLC · p. 8, Figure 11 · p. 8
  • Code 1 has a noticeable error floor with three or six reads, while Code 2 avoids the cited small absorbing sets because it has no degree-three variable nodes.The (4,2) absorbing set can be avoided by precluding degree-three variable nodes.Source: A. MMI Quantization for MLC · p. 8, Figure 12 · p. 8

B. A More Realistic Model

The realistic six-month retention model has asymmetric conditional noise distributions, so its read-channel transition probabilities and MI are computed numerically. The resulting MI optimization uses numerical search in the locally quasi-concave region of interest.

B. A More Realistic Model

  • The six-month retention model yields four conditional threshold-voltage distributions with substantially different variances, the lowest-threshold distribution having the largest variance.The six word-line voltages shown are those that maximize MI for this noise model.Source: B. A More Realistic Model · p. 8, Figure 13 · p. 9
  • Because the retention noise model is asymmetric, transition probabilities must be computed numerically to calculate MI.The channel is treated as a four-input, seven-output discrete memoryless channel.Source: B. A More Realistic Model · pp. 8–9
  • Although MI is generally not quasi-concave in the word-line voltages, it is locally quasi-concave in the region of interest, permitting numerical search methods.The authors cite careful bisection search or other quasi-convex optimization techniques.Source: B. A More Realistic Model · p. 9

C. The Constant-Ratio Method

For Code 2, the MI-maximizing constant-ratio choice also gives the lowest FER in both the Gaussian and retention models. The constraint preserves performance close to unconstrained MMI while reducing the optimization search space.

C. The Constant-Ratio Method

  • The constant-ratio method sets secondary thresholds by fixing the ratio of the largest and second-largest conditional noise densities, reducing MMI optimization to a search over R.Choosing R to maximize MI makes the constrained MI function quasi-concave in the region of interest for the symmetric Gaussian channel.Source: C. The Constant-Ratio Method · p. 9
  • For both channel models, MI-maximizing constant-ratio quantization provides essentially the same MI as unconstrained MMI optimization.The Gaussian and retention models have similar MI-versus-R behavior, though the retention curve is not strictly concave.Source: C. The Constant-Ratio Method · pp. 9–10
  • For Code 2, the R that maximizes MI also gives the lowest FER across the Gaussian and six-month retention models.The figure compares MI and FER against R for both models.Source: Figure 14 · p. 9, C. The Constant-Ratio Method · p. 10
  • The constant-ratio constraint’s lowest FER is essentially the same as unconstrained MMI, although an approximately 0.01-bit MI variation corresponds to more than an order-of-magnitude FER difference.The figure reports MMI of 1.885 bits for both models and shows their MMI-quantized FER values.Source: Figure 14 · p. 9, C. The Constant-Ratio Method · p. 10

D. MMI Optimization Thwarted by Small Absorbing Sets

For Code 1, the lowest FER occurs at a constant-ratio value with lower MI than the MI-maximizing choice. The paper attributes this result to absorbing-set traps that favor a wider erasure region.

D. MMI Optimization Thwarted by Small Absorbing Sets

  • The paper cautions that poorly designed codes can perform best with quantization that does not maximize channel MI.The result contrasts with the well-designed-code example discussed immediately before this case.Source: D. MMI Optimization Thwarted by Small… · p. 10
  • For Code 1, the lowest FER occurs at R = 15, which provides less MI than R = 7.This is the example in which the code’s best FER quantization does not maximize channel MI.Source: D. MMI Optimization Thwarted by Small… · p. 10
  • Numerous small absorbing sets can turn a few hard-decoded errors into uncorrectable problems, favoring a wider erasure region than capacity would favor.The wider region minimizes hard-decoded errors that trigger the absorbing sets.Source: D. MMI Optimization Thwarted by Small… · p. 10

E. Simulation Results for Retention Model

In the retention model, Code 2 outperforms Code 1 at both three and six reads, while MMI quantization matches the lowest-FER choice for Code 2 but not for Code 1. The conclusion is that a few extra reads can capture most of the benefit of enhanced precision, with code quality and decoder representation remaining important qualifications.

E. Simulation Results for Retention Model

  • Code 2 outperforms Code 1 with both three and six reads, and its unconstrained-MMI and R = 7 FER curves are indistinguishable.R = 7 maximizes mutual information and minimizes FER for Code 2.Source: E. Simulation Results for Retention… · p. 10
  • Allowing asymmetric thresholds for the non-symmetric retention model produced insignificant gains in the simulations.The three-read quantization shown for the retention model is standard hard decoding for four-level MLC.Source: E. Simulation Results for Retention… · p. 10
  • For Code 1 with six reads, R = 15 yields the lowest FER even though it provides less mutual information than R = 7.The MMI quantization performs slightly worse than R = 15 for this code.Source: E. Simulation Results for Retention… · p. 11

VI. CONCLUSION

  • A few additional reads harvest most of the enhanced-precision benefit, and constant-ratio optimization simplifies word-line-voltage selection without noticeable performance loss.The paper also reports that small amounts of soft information improve LDPC performance and give a clear advantage over conventional BCH codes.Source: §VI · p. 11
  • For well-designed LDPC codes, MMI quantization can minimize FER, but high-error-floor codes may have a different lowest-FER quantization and code performance can vary across precision levels.The conclusion identifies degree-distribution mismatch across full-precision and quantized settings, as well as more important absorbing sets at lower precision.Source: §VI · p. 11
  • The simulations quantize channel information but retain floating-point messages between decoder nodes, leaving limited-precision decoder representations for future investigation.The paper proposes studying decoder-message precision together with the limited-precision channel information available in Flash memory.Source: §VI · p. 11
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