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

Roswitha Hofer, Kosuke Suzuki

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 $2$.

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

Abstract

We give a characterization of all matrices which generate a -net in base and a characterization of all matrices which generate a -sequence in base .

Paper Structure

This paper contains 3 sections, 12 theorems, 7 equations.

Table of Contents

  1. Introduction and main results
  2. Auxiliary results
  3. Proof of Theorem \ref{['thm:char-0m3met-general']} and \ref{['thm:char-02seq']}

Key Result

Theorem 1.1

Let $m \geq 1$ be an integer and $A,\,B,\,C \in \mathbb{F}_2^{m \times m}$. Then the following are equivalent.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Kajiura
  • Lemma 2.2
  • proof
  • Lemma 2.3: Niederreiter1992rng
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 12 more