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Some interesting behaviors of good lattice point sets

Abstract Good lattice point (GLP) sets are sets of points that are uniformly distributed over the domain of interest and thus have good space-filling property. GLP sets are an important kind of points for many applications, such as multidimensional quadrature, simulation, computer experiments, quasi-Monte Carlo techniques and design of experiments. It is a significant issue to study the behaviors of GLP sets. For instance, most real-life applications require that the columns of a GLP set are independent, i.e., the GLP set has full rank. If one or more columns are linearly dependent, there will be more confounding among the main effects and interactions in statistical models and also we can not find the least squares estimation of regression coefficients in the linear regression model. The problem of finding the rank of the GLP set remained unsolved since its inception in 1959. It is desirable to put the first stone for solving this significant problem. Through theoretical justifications, this paper gives some interesting behaviors of the generating vector of any GLP set and shows the independence among its rows and columns by presenting some analytic linkages among these rows and columns. The results of this paper are used not only as a benchmark for constructing full rank GLP sets, but also in various applications of GLP sets.