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

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 $B^3=I$.