$L_2$ discrepancy of symmetrized generalized Hammersley point sets in base $b$

TL;DR

This paper investigates the $L_2$ discrepancy of symmetrized, scrambled $b$-adic Hammersley point sets, providing exact formulas for special permutations and identifying optimal permutations for bases 2 to 27.

Contribution

It combines symmetrization and digit scrambling techniques, derives exact $L_2$ discrepancy formulas, and finds permutations that minimize discrepancy across multiple bases.

Findings

Exact $L_2$ discrepancy formulas for special permutations.

Identification of permutations with lowest $L_2$ discrepancy for bases 2 to 27.

Confirmation that symmetrization and scrambling improve discrepancy properties.

Abstract

Two popular and often applied methods to obtain two-dimensional point sets with the optimal order of $L_p$ discrepancy are digit scrambling and symmetrization. In this paper we combine these two techniques and symmetrize $b$-adic Hammersley point sets scrambled with arbitrary permutations. It is already known that these modifications indeed assure that the $L_p$ discrepancy is of optimal order $\mathcal{O}\left(\sqrt{\log{N}}/N\right)$ for $p\in [1,\infty)$ in contrast to the classical Hammersley point set. We prove an exact formula for the $L_2$ discrepancy of these point sets for special permutations. We also present the permutations which lead to the lowest $L_2$ discrepancy for every base $b\in\{2,\dots,27\}$ by employing computer search algorithms.