A generalization of short-period Tausworthe generators and its application to Markov chain quasi-Monte Carlo
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A one-dimensional sequence u_0, u_1, u_2, …∈ [0, 1) is said to be completely uniformly distributed (CUD) if overlapping s-blocks (u_i, u_i+1, … , u_i+s-1), i = 0, 1, 2, …, are uniformly distributed for every dimension s ≥ 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 𝔽_2 that approximate CUD sequences by running for the entire period. In this paper, we generalize a search algorithm over 𝔽_2 to that over arbitrary finite fields 𝔽_b with b elements and conduct a search for Tausworthe generators over 𝔽_b with t-values zero (i.e., optimal) for dimension s = 3 and small for s ≥ 4, especially in the case where b = 3, 4, and 5. We provide a parameter table of Tausworthe generators over 𝔽_4, and report a comparison between our new generators over 𝔽_4 and existing generators over 𝔽_2 in numerical examples using Markov chain QMC.
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