A search for short-period Tausworthe generators over F b with application to Markov chain quasi-Monte Carlo

Abstract

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 [A table of short-period Tausworthe generators for Markov chain quasi-Monte Carlo. J Comput Appl Math. 2021;384:Paper No. 113136, 12.] focussed 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.