Exact Projection Quality of OneTwo Sobol' Sequences at 65,536 Points

Low-dimensional projection quality is a practical bottleneck for Sobol' sequences used in quasi-Monte Carlo integration and computer graphics. For a digital net with 2^16 = 65,536 points, the t-value measures exact equidistribution in dyadic boxes, with smaller values indicating stronger uniformity. We determine the exact m = 16 census for all 345 pair-aligned four-dimensional blocks in the released 692-dimensional OneTwo construction: 9 blocks have t = 4, and the remaining 336 have t = 5. For dimensions 25–28, determinant-one minors prove the upper bound, while a unique row dependency with mask 0xD2F yields an explicit empty dyadic box for the lower bound. Exact computations over F_2 agree across three independent rank routes, direct enumeration checks all 65,536 points, and a Lean companion checks the finite worked-block encoding. The worked certificate is self-contained; reproducing the complete census additionally uses the exact upstream direction-number table identified by a pinned public commit and file hashes.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-08-14
DOI
https://doi.org/10.5281/zenodo.21925582
Primary Topic
Mathematical Approximation and Integration
Type
preprint
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preprint

Exact Projection Quality of OneTwo Sobol' Sequences at 65,536 Points

Lennart Rudolph
Zenodo (CERN European Organization for Nuclear Research)
Mathematical Approximation and Integration
preprint

Exact Projection Quality of OneTwo Sobol' Sequences at 65,536 Points

Lennart Rudolph
preprint en

Abstract

Low-dimensional projection quality is a practical bottleneck for Sobol' sequences used in quasi-Monte Carlo integration and computer graphics. For a digital net with 2^16 = 65,536 points, the t-value measures exact equidistribution in dyadic boxes, with smaller values indicating stronger uniformity. We determine the exact m = 16 census for all 345 pair-aligned four-dimensional blocks in the released 692-dimensional OneTwo construction: 9 blocks have t = 4, and the remaining 336 have t = 5. For dimensions 25–28, determinant-one minors prove the upper bound, while a unique row dependency with mask 0xD2F yields an explicit empty dyadic box for the lower bound. Exact computations over F_2 agree across three independent rank routes, direct enumeration checks all 65,536 points, and a Lean companion checks the finite worked-block encoding. The worked certificate is self-contained; reproducing the complete census additionally uses the exact upstream direction-number table identified by a pinned public commit and file hashes.

Zenodo (CERN European Organization for Nuclear Research)
Mathematical Approximation and Integration
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Exact Projection Quality of OneTwo Sobol' Sequences at 65,536 Points — Lennart Rudolph · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS