tklds — High-Dimensional Sobol’ Sequences for Python

tklds (Tenokonda Low Discrepancy Sequences) is a Python package and direction-number dataset for generating, scrambling, and evaluating high-dimensional Sobol’ sequences for quasi-Monte Carlo (QMC) simulation.

The package includes the tkrg-a-ap5 direction-number set for 50,000 dimensions. The construction satisfies Property A in every supported dimension and Property A′ on every block of five adjacent dimensions. It also includes the new-joe-kuo-6.21201 direction numbers for reference and comparison.

The library provides:

• batch and iterative Sobol’ point generation;
• a SciPy-compatible SobolEngine with optional scrambling;
• the tkrg-a-ap5 and new-joe-kuo-6.21201 direction-number datasets;
• diagnostics for high-dimensional sequence behaviour;
• worked notebooks for integration, stochastic processes, pricing, sensitivities, and risk; and
• reproducibility material for the accompanying 2026 paper.

Scope note: Direction-number performance depends on the integrand, coordinate ordering, path construction, scrambling method, and sample size. The supplied research benchmarks identify material advantages for the tested high-dimensional path-simulation regimes; they do not establish that one direction-number set dominates every QMC problem.

Install

From PyPI

python -m pip install tklds

From a source checkout

git clone https://github.com/TENOKONDA/tklds.git
cd tklds
python -m pip install .

Editable development installation

git clone https://github.com/TENOKONDA/tklds.git
cd tklds
python -m pip install -e ".[examples,test]"

tklds requires Python 3.10 or later. The examples extra installs the packages used by the notebooks; the test extra installs the proposed test runner and coverage tooling.

Quick start

Generate a batch of Sobol’ points

from tklds.constant import SequenceNum
from tklds.interface.generators import generate_lds_rvs

points = generate_lds_rvs(
    sequence=SequenceNum.TKRG_A_AP5,
    n=1_024,
    d=32,
    skip=0,
)

print(points.shape)  # (1024, 32)

Use the SciPy-compatible engine

from tklds.constant import SequenceNum
from tklds.interface.generators import create_sobol_lds_engine

engine = create_sobol_lds_engine(
    sequence=SequenceNum.TKRG_A_AP5,
    d=256,
    scramble=True,
    seed=1234,
)

# 2**10 points. Power-of-two sample sizes preserve the natural
# balance properties of a Sobol’ digital net.
points = engine.random_base2(m=10)
print(points.shape)  # (1024, 256)

Generate points iteratively

from tklds.constant import SequenceNum
from tklds.interface.generators import create_iterative_lds_generator

sequence = create_iterative_lds_generator(
    sequence=SequenceNum.TKRG_A_AP5,
    d=8,
)

first_batch = sequence.rvs(size=(16, 8))
second_batch = sequence.rvs(size=(16, 8))

Available direction-number sets

The raw direction-number files are distributed with the package under tklds/direction_numbers.

Choosing a generation method

Use generate_lds_rvs for a direct array of points, create_iterative_lds_generator when points are consumed in successive batches, and create_sobol_lds_engine when interoperability with the SciPy QMC interface is useful.

For deterministic Sobol’ experiments, use scramble=False. For replicate-based error estimation, use independent scrambled engines with controlled seeds. Where the balance properties of the underlying digital net matter, prefer n = 2**m and random_base2(m).

Examples and reproducibility

The repository separates introductory examples from paper-specific reproduction material:

• notebooks/01_quickstart.ipynb — package introduction;
• notebooks/02_integral.ipynb — numerical integration;
• notebooks/05_spurious_variance.ipynb — transformed-coordinate dependence diagnostic;
• notebooks/06_brownian_motion.ipynb — process simulation;
• notebooks/07_sobol_engine_examples.ipynb — SciPy-compatible engine usage; and
• notebooks/paper_notebooks — figures and benchmarks associated with the 2026 paper.

Notebook dependencies are installed with:

python -m pip install -e ".[examples]"

Research basis

The current technical reference is:

Andres Oliva Denis, James Wheeldon, Ilya Manyakin, and Adrien Papaioannou. Sobol Direction Numbers for Quasi-Monte Carlo Simulation: Construction, Pathwise Dependence Diagnostics, and Financial Pricing Benchmarks. Tenokonda UK, 30 April 2026. SSRN abstract 7040539.

The paper introduces tkrg-a-ap5, explains its construction over GF(2), develops a pathwise dependence diagnostic, and reports deterministic pricing and risk benchmarks under standard chronological discretisation.

• Read the paper on SSRN
• Citation instructions
• Machine-readable citation
• BibTeX entries

Testing

The repository contains unit tests for the public generator interfaces, iterative generation, the SciPy-compatible Sobol engine, numerical integration utilities, stochastic-process components, and the spurious-variance diagnostic.

Run the current suite from a source checkout with:

python -m unittest discover -s tklds/tests -p "test_*.py"

Automated pull-request and release-gating workflows are a high-priority follow-on item. Until those workflows are added, maintainers should run the complete suite manually before tagging a release.

Project links

• Source code
• PyPI package
• Issue tracker
• Release history
• Tenokonda

Licence

tklds is distributed under the BSD 3-Clause License.