Characterization of matrices $B$ such that $(I,B,B^2)$ generates a digital net with $t$-value zero

TL;DR

This paper characterizes matrices B over GF(2) that generate 3-dimensional digital nets with zero t-value, showing such B must satisfy B^3=I, thus providing a clear algebraic condition for optimal digital net construction.

Contribution

It provides a complete characterization of matrices B that produce digital nets with t-value zero, linking algebraic properties of B to net quality.

Findings

Matrices B with B^3=I generate digital nets with t-value zero.

The characterization links the algebraic condition B^3=I to the net's t-value.

The result simplifies the identification of optimal digital nets in 3D over GF(2).

Abstract

We study $3$-dimensional digital nets over $\mathbb{F}_2$ generated by matrices $(I,B,B^2)$ where $I$ is the identity matrix and $B$ is a square matrix. We give a characterization of $B$ for which the $t$-value of the digital net is $0$. As a corollary, we prove that such $B$ satisfies $B^3=I$.