On the intriguing search for good permutations

TL;DR

This paper surveys Henri Faure's four-decade research on permutations that generate low-discrepancy generalized van der Corput sequences, highlighting different approaches, open questions, and future directions.

Contribution

It provides a comprehensive overview of methods and open problems in the search for permutations producing low-discrepancy sequences, inspired by Henri Faure's work.

Findings

Comparison of permutation-based and sequence-based approaches

Collection of open questions in low-discrepancy sequence construction

Extension and sharpening of existing open problems

Abstract

The intriguing search for permutations that generate generalised van der Corput sequences with exceptionally small discrepancy forms an important part of the research work of Henri Faure. On the occasion of Henri's 80th birthday we aim to survey (some of) his contributions over the last four decades which considerably improved our understanding of one-dimensional van der Corput sequences and inspired a lot of related work. We recall and compare the different approaches in the search for generalised van der Corput sequences with low discrepancy, i.e., using a single generating permutation versus using a sequence of permutations. Throughout, we collect, sharpen and extend open questions which all stem from the extensive work of Henri and his coworkers and which will hopefully inspire more work in the future.