Analysis and Synthesis of Digital Dyadic Sequences

Abdalla G. M. Ahmed, Markus Hadwiger, Mikhail Skopenkov and Peter Wonka

Analysis and Synthesis of Digital Dyadic Sequences
arXiv:2306.06925 , 2023

We explore the space of matrix-generated (0, m, 2)-nets and (0, 2)-sequences in base 2, also known as digital dyadic nets and sequences. We provide a complete characterization of the design space and count the possible number of constructions with and without considering possible reorderings of the point set. Based on this analysis, we then show that every digital dyadic net can be reordered into a sequence, together with a corresponding algorithm. Finally, we present a novel family of self-similar digital dyadic sequences, to be named xi-sequences, that spans a subspace with fewer degrees of freedom. Those xi-sequences are extremely efficient to sample and compute, and we demonstrate their advantages over the classic Sobol (0, 2)-sequence.

  title = {Analysis and Synthesis of Digital Dyadic Sequences},
  author = {Ahmed, Abdalla G. M. and Hadwiger, Markus and Skopenkov, Mikhail and Wonka, Peter},
  journal = {arXiv e-prints},
  archivePrefix = "arXiv",
  eprint = {2306.06925},
  year = {2023}