# On the Classification of LS-Sequences

## TL;DR

This paper investigates the classification of LS-sequences, demonstrating that only specific cases correspond to known low discrepancy sequences and providing improved discrepancy bounds for certain cases.

## Contribution

It proves that LS-sequences with S=1 relate to Kronecker sequences and that for S≥2 they are distinct, also offering improved discrepancy bounds for S=1.

## Key findings

- LS-sequences with S=0 are van der Corput sequences
- S=1 sequences relate to two-sided Kronecker sequences
- For S≥2, LS-sequences are neither van der Corput nor Kronecker

## Abstract

This paper adresses the question whether the $LS$-sequences constructed by Carbone yield indeed a new family of low discrepancy sequences. While it is well known that the case $S=0$ corresponds to van der Corput sequences, we prove here that the case $S=1$ can be traced back to two-sided Kronecker sequences and moreover that for $S \geq 2$ none of these two types occurs anymore. In addition, our approach allows for an improved discrepancy bound for $S=1$ and $L$ arbitrary.