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

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