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A one-dimensional sequence $u_0, u_1, u_2, \ldots \in [0, 1)$ is said to becompletely uniformly distributed (CUD) if overlapping $s$-blocks $(u_i,u_{i+1}, \ldots , u_{i+s-1})$, $i = 0, 1, 2, \ldots$, are uniformly distributedfor every dimension $s \geq 1$. This concept naturally arises in Markov chainquasi-Monte Carlo (QMC). However, the definition of CUD sequences is notconstructive, and thus there remains the problem of how to implement the Markovchain QMC algorithm in practice. Harase (2021) focused on the $t$-value, whichis a measure of uniformity widely used in the study of QMC, and implementedshort-period Tausworthe generators (i.e., linear feedback shift registergenerators) over the two-element field $\mathbb{F}_2$ that approximate CUDsequences by running for the entire period. In this paper, we generalize asearch algorithm over $\mathbb{F}_2$ to that over arbitrary finite fields$\mathbb{F}_b$ with $b$ elements and conduct a search for Tausworthe generatorsover $\mathbb{F}_b$ with $t$-values zero (i.e., optimal) for dimension $s = 3$and small for $s \geq 4$, especially in the case where $b = 3, 4$, and $5$. Weprovide a parameter table of Tausworthe generators over $\mathbb{F}_4$, andreport a comparison between our new generators over $\mathbb{F}_4$ and existinggenerators over $\mathbb{F}_2$ in numerical examples using Markov chain QMC.

A search for short-period Tausworthe generators over $\mathbb{F}_b$ with application to Markov chain quasi-Monte Carlo