On the Discrepancy of Two Families of Permuted Van der Corput Sequences

Abstract

A permuted van der Corput sequence $S_b^\sigma$ in base b is a one-dimensional, infinite sequence of real numbers in the interval [0, 1), generation of which involves a permutation σ of the set {0, 1,..., b − 1}. These sequences are known to have low discrepancy DN, i.e. $t\left({S_b^\sigma } \right): = {\rm{lim}}\,{\rm{sup}}_{N \to \infty } D_N \left({S_b^\sigma } \right)/{\rm{log}}\,N$ is finite. Restricting to prime bases p we present two families of generating permutations. We describe their elements as polynomials over finite fields 𝔽p in an explicit way. We use this characterization to obtain bounds for $t\left({S_p^\sigma } \right)$ for permutations σ in these families. We determine the best permutations in our first family and show that all permutations of the second family improve the distribution behavior of classical van der Corput sequences in the sense that $t\left({S_p^\sigma } \right) < t\left({S_p^{id} } \right)$ .

On the Intriguing Search for Good Permutations

“…The algorithm of Faure is the main motivation for the results of Pausinger & Topuzoglu [51] presented (among other things) in this section. One disadvantage of Faure's algorithm is that we only get one permutation in a given base, and constructing this permutation requires the construction of permutations in smaller bases.…”

“…One disadvantage of Faure's algorithm is that we only get one permutation in a given base, and constructing this permutation requires the construction of permutations in smaller bases. In [51] the authors aim to give discrepancy bounds for sequences generated from structurally similar permutations in a given (prime) base p. The advantage of restricting to prime bases p is that finite fields F p of p elements are polynomially complete. This means that any self map, and in particular any permutation of F p , can be expressed as a polynomial over F p .…”

“…Interestingly, it was observed in [51] that for every permutation π there exists a permutation τ such that π(x) = τ (x) for all x ∈ F p \ {X 1 , X 2 }; i.e., we can attach p − 1 permutations τ to each permutation π.…”

“…Let σ 60 = (0, 15,30,40,2,48,20,35,8,52,23,43,12,26,55,4,32,45,17,37,6,50,28,10,57,21,41,13,33,54,1,25,46,18,38,5,49,29,9,58,22,42,14,34,53,3,27,47,16,36,7,51,19,44,…”

Estimation of Volume Using the Nucleator and Lattice Points

“…The Gaussian particle is very smooth whereas the computer reconstruction of the rat brain has edges (and as such may not fully qualify as a nice particle). We use the following point sets in the unit square to generate direction vectors u ∈ S 2 for the rays in our volume estimation: 1) Lattice points -for a given N we choose the generating vector (1, q), with 1 ≤ q ≤ N − 1, such that q, N are coprime and the resulting lattice has the smallest discrepancy among all possible such lattices with N points; for details we refer to (Pausinger and Topuzoglu, 2018). Note that discrepancy is a standard tool from uniform distribution theory to assess the distribution quality of a point set used in numerical integration.…”