LDData: Low-Discrepancy Generating Vectors and Matrices

A curated collection of low-discrepancy point set parameters including lattice rules, digital nets, polynomial lattice rules, Sobol' nets, and RQMC randomizations. This dataset enables reproducible research and high-performance Quasi–Monte Carlo (QMC) and Randomized QMC (RQMC) simulation.

The LDData repository provides standard text-based formats for specifying structures used in QMC point generation across arbitrarily high dimensions.

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Dataset Summary

LDData is a dataset of structured parameter files defining:

• Rank-1 lattice rules
• Base-$b$ digital nets
• Polynomial lattice rules
• Sobol' and Sobol–Joe-Kuo sequences
• Various randomizations (shift modulo 1, digital shifts, nested uniform scrambles, left matrix scrambles)

Each file type follows a simple textual standard to ensure:

• Human readability
• Language-agnostic parsing
• Long-term reproducibility
• Extensibility to thousands of dimensions

The dataset is motivated by the need for standardized, compact, transparent formats in our QMC research and software.

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Motivation

Many QMC constructions appear across scattered software packages, papers, or custom formats. LDData brings these formats together into a consistent, unified, machine-readable repository for:

• Researchers developing new QMC methods
• Practitioners needing high-dimensional low-discrepancy point sets
• Developers of simulation libraries such as SSJ, QMCPy, and LatNet Builder

This dataset is linked to the research works described in the Citation section below. For detailed technical specifications and implementation details, see LD_DATA.md

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Supported Tasks and Applications

✔️ Quasi-Monte Carlo (QMC)

Generate deterministic point sets with excellent equidistribution.

✔️ Randomized QMC (RQMC)

Use the included randomizations for variance estimation:

• Digital shifts
• Nested uniform scrambles
• Left-matrix scrambles

✔️ High-dimensional Integration and Simulation

Used in:

• Bayesian computation
• Option pricing
• High-dimensional PDE solvers
• Uncertainty quantification
• Graphics and rendering research
• Machine learning sampling methods

✔️ Benchmarking

Standard formats help evaluate new constructions against established ones.

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Features

• Simple .txt formats with one line per dimension
• Optional human-readable comments starting with #
• No binary encoding or word-size assumptions
• Supports extremely high dimensions (10,000+)
• Extensible constructions (e.g., Sobol or embedded nets)
• All formats interoperable with QMC software (SSJ, QMCPy, LatNet Builder)

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How to Use the Dataset

Load files directly from Hugging Face

from datasets import load_dataset

ds = load_dataset("QMCSoftware/LDData")

All data files are preserved in their directory structure and can be accessed using:

ds["train"]  # or ds['default']

Typical workflow

1. Read a parameter file (e.g. lattice_8d.txt)
2. Parse header (# lattice, dimensions, n, etc.)
3. Parse one line per dimension for the generating vector or matrices
4. Construct QMC point generator in your preferred library

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Dataset Structure

The dataset includes multiple categories of files:

🔹 lattice

Rank-1 lattice generating vectors:

• Header: # lattice
• Parameters:
  • Number of dimensions s
  • Number of points n
  • s lines of generating vector coefficients

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🔹 dnet

General digital nets in base b:

• Header: # dnet
• Parameters:
  • Base b
  • Dimensions s
  • Columns k
  • Rows r
• Then s lines representing generating matrices

Efficient for high-dimensional digital nets.

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🔹 plattice

Polynomial lattice rules:

• Compact format using integer-encoded polynomials
• Base b, dimension s, polynomial degree k, and generating polynomials

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🔹 sobol and soboljk

Parameters for Sobol' sequences:

• soboljk: Joe & Kuo format with primitive polynomials and direction numbers
• sobol: Simplified direction-number only format

Used widely in QMC applications.

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🔹 Randomization formats

Includes:

• shiftmod1: Shift modulo 1
• dshift: Digital shift in base b
• nuscramble: Nested uniform scramble
• lmscramble: Left matrix scramble

All formats are text-based and reproducible.

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Example: Parsing a Lattice Rule File

Example file:

# lattice
8
65536
1 
19463
17213
5895
14865
31925
30921
26671

Python pseudo-code:

with open("lattice_8d.txt") as f:
    lines = [l for l in f.readlines() if not l.startswith("#")]

s = int(lines[0])
n = int(lines[1])
a = [int(x) for x in lines[2:2+s]]

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File Naming Recommendations

To support discoverability and consistent tooling:

• All files begin with their keyword (lattice_, dnet_, sobol_, etc.)
• Headers contain:
  • Construction method
  • Figure of merit (FOM)
  • Weights
  • Embedded range (if applicable)
• Comments allowed in headers only

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References

This dataset incorporates formats and ideas from foundational work in QMC:

• Bratley & Fox (1988)
• Joe & Kuo (2008)
• L’Ecuyer (2016)
• Goda & Dick (2015)
• Nuyens (2020)
• And others listed in the detailed specification LD_DATA.md.

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Citation

If you use LDData in academic work, please cite:

@article{sorokin.2025.ld_randomizations_ho_nets_fast_kernel_mats,
  title               = {{QMCPy}: a {P}ython software for randomized low-discrepancy sequences, quasi-{M}onte {C}arlo, and fast kernel methods},
  author              = {Aleksei G. Sorokin},
  year                = {2025},
  journal             = {ArXiv preprint},
  volume              = {abs/2502.14256},
  url                 = {https://arxiv.org/abs/2502.14256},
}


@inproceedings{choi.QMC_software,
  title               = {Quasi-{M}onte {C}arlo software},
  author              = {Choi, Sou-Cheng T. and Hickernell, Fred J. and Rathinavel, Jagadeeswaran and McCourt, Michael J. and Sorokin, Aleksei G.},
  year                = {2022},
  booktitle           = {{M}onte {C}arlo and Quasi-{M}onte {C}arlo Methods 2020},
  publisher           = {Springer International Publishing},
  address             = {Cham},
  pages               = {23--47},
  isbn                = {978-3-030-98319-2},
  editor              = {Keller, Alexander},
}

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License

Apache 2 License.
See LICENSE file for details.

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Acknowledgements

This dataset is developed and maintained by:

• QMCSoftware team
• Contributors to QMCPy, SSJ, and LatNet Builder
• Community contributions from QMC & RQMC researchers

Special thanks to researchers providing widely used generating vectors and direction numbers used throughout the scientific computing community.