On the (WGb,φ; α)− diaphony of the nets of Halton- Zaremba constructed over finite groups

Abstract

In the present paper, the authors introduce an arithmetic based on finite groups with respect to arbitrary bijections. This algebraic background is used to construct the function system W_Gb,φb of the Walsh functions over the set G_b of groups with respect to the set φb of bijections. The developed algebraic base is also used to introduce a wide class of two-dimensional nets Gb,φbZ κ,µ b,ν of type of Halton-Zaremba. Four concrete nets of this class are constructed and graphically illustrated. The so-called (W_Gb,φ; α)−diaphony is applied as an appropriate tool for studying the nets of the introduced class. An exact formula for the (W_Gb,φ; α)−diaphony of the nets of class Gb,φbZ κ,µ b,ν is presented. This formula allows us to show the influence of the vector α on the exact order of the (W_Gb,φ; α)−diaphony of these nets.