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

A permuted van der Corput sequence Sbσ$S_b^\sigma$ 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(Sbσ):=lim supN→∞DN(Sbσ)/log N$t\left({S_b^\sigma } \right): = {\rm{lim}}\,{\rm{sup}}_{N \to \infty } D_N \left({S_b^\sigma } \right)/{\rm{log}}\,N$ 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(Spσ)$t\left({S_p^\sigma } \right)$ 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(Spσ)<t(Spid)$t\left({S_p^\sigma } \right) < t\left({S_p^{id} } \right)$.