  A one-dimensional sequence $u_0, u_1, u_2, \ldots \in [0, 1)$ is said to be
completely uniformly distributed (CUD) if overlapping $s$-blocks $(u_i,
u_{i+1}, \ldots , u_{i+s-1})$, $i = 0, 1, 2, \ldots$, are uniformly distributed
for every dimension $s \geq 1$. This concept naturally arises in Markov chain
quasi-Monte Carlo (QMC). However, the definition of CUD sequences is not
constructive, and thus there remains the problem of how to implement the Markov
chain QMC algorithm in practice. Harase (2021) focused on the $t$-value, which
is a measure of uniformity widely used in the study of QMC, and implemented
short-period Tausworthe generators (i.e., linear feedback shift register
generators) over the two-element field $\mathbb{F}_2$ that approximate CUD
sequences by running for the entire period. In this paper, we generalize a
search algorithm over $\mathbb{F}_2$ to that over arbitrary finite fields
$\mathbb{F}_b$ with $b$ elements and conduct a search for Tausworthe generators
over $\mathbb{F}_b$ 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 $5$. We
provide a parameter table of Tausworthe generators over $\mathbb{F}_4$, and
report a comparison between our new generators over $\mathbb{F}_4$ and existing
generators over $\mathbb{F}_2$ in numerical examples using Markov chain QMC.
