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$u_0, u_1, u_2, \ldots \in [0, 1)$被称为完全均匀分布（CUD），如果对于任意维度$s \geq 1$，其重叠$s$块$(u_i, u_{i+1}, \ldots , u_{i+s-1})$（$i = 0, 1, 2, \ldots$）均匀分布。该概念自然出现在马氏链拟蒙特卡洛（QMC）方法中。然而CUD序列的定义不具有构造性，因此实际如何实现马氏链QMC算法仍存在问题。Harase（2021）聚焦于QMC研究中广泛使用的均匀性度量指标$t$值，通过运行完整周期实现了近似CUD序列的二元域$\mathbb{F}_2$上短周期Tausworthe生成器（即线性反馈移位寄存器生成器）。本文将$\mathbb{F}_2$上的搜索算法推广至任意$b$元有限域$\mathbb{F}_b$，并对$\mathbb{F}_b$上Tausworthe生成器进行搜索，要求其在维度$s = 3$时$t$值为零（即最优），在$s \geq 4$时$t$值较小，特别针对$b = 3, 4, 5$的情形。我们给出了$\mathbb{F}_4$上Tausworthe生成器的参数表，并通过马氏链QMC数值算例报告了$\mathbb{F}_4$上新型生成器与现有$\mathbb{F}_2$上生成器的性能比较。