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.

翻译：一维序列 $u_0, u_1, u_2, \ldots \in [0, 1)$ 若其重叠的 $s$ 块 $(u_i, u_{i+1}, \ldots , u_{i+s-1})$（$i = 0, 1, 2, \ldots$）对每个维度 $s \geq 1$ 都是均匀分布的，则称该序列为完全均匀分布（CUD）序列。这一概念自然出现在马尔可夫链拟蒙特卡洛（QMC）中。然而，CUD序列的定义并非构造性的，因此在实际中如何实现马尔可夫链QMC算法仍是一个问题。Harase（2021）聚焦于$t$值（QMC研究中广泛使用的均匀性度量），并实现了在二元域$\mathbb{F}_2$上运行整个周期以近似CUD序列的短周期Tausworthe生成器（即线性反馈移位寄存器生成器）。在本文中，我们将$\mathbb{F}_2$上的搜索算法推广到任意有限域$\mathbb{F}_b$（含$b$个元素），并在$\mathbb{F}_b$上搜索Tausworthe生成器，使其$t$值在维度$s=3$时为零（即最优），在$s\geq4$时较小，特别关注$b=3,4,5$的情形。我们提供了$\mathbb{F}_4$上Tausworthe生成器的参数表，并通过使用马尔可夫链QMC的数值算例，比较了$\mathbb{F}_4$上的新生成器与$\mathbb{F}_2$上的现有生成器。