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：若一维序列$u_0, u_1, u_2, \ldots \in [0, 1)$满足：对任意维度$s \geq 1$，其重叠$s$-块$(u_i, u_{i+1}, \ldots , u_{i+s-1})$（$i = 0, 1, 2, \ldots$）均为均匀分布，则称该序列为完全均匀分布（CUD）。该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），并实现了二元域$\mathbb{F}_2$上的短周期Tausworthe生成器（即线性反馈移位寄存器生成器），通过运行整个周期来近似CUD序列。本文将该搜索算法从$\mathbb{F}_2$推广到具有$b$个元素的任意有限域$\mathbb{F}_b$，并在$\mathbb{F}_b$上搜索$t$值为零（即最优）的Tausworthe生成器（针对维度$s=3$）以及$s\geq 4$时$t$值较小的生成器，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，并在使用马尔可夫链QMC的数值算例中，比较了本研究所提$\mathbb{F}_4$上的新生成器与现有$\mathbb{F}_2$上生成器的性能。