Characterization of digital $(0,m,3)$-nets and digital $(0,2)$-sequences in base $2$

TL;DR

This paper provides a complete characterization of matrices over the finite field _2 that generate digital _2-based (0,m,3)-nets and _2-based (0,2)-sequences, advancing the understanding of their structure.

Contribution

It offers a comprehensive description of all matrices producing these digital nets and sequences in base 2, filling a gap in the theoretical understanding.

Findings

Characterization of matrices generating (0,m,3)-nets in base 2.

Characterization of matrices generating (0,2)-sequences in base 2.

Provides criteria for matrix construction of digital nets and sequences.

Abstract

We give a characterization of all matrices $A,B,C \in \mathbb{F}_{2}^{m \times m}$ which generate a $(0,m,3)$-net in base $2$ and a characterization of all matrices $B,C\in\mathbb{F}_{2}^{\mathbb{N}\times\mathbb{N}}$ which generate a $(0,2)$-sequence in base $2$.