Further Study On Good Lattice Point Design

Uniform design is an important type of experimental design.It has been applied to medical and health care,biotechnology,space technology,material chemical industry and other fields of social production and life.During this process,the corresponding theory of uniform design has been proposed,such as the definition of different uniformity measures,the discussion of properties under different uniformity measures,and the construction of uniform design.Owing to the flexible numbers of runs,uniform design is quite popular in practical applications.A theory shows that when constructing uniform design,we should choose the design with high factor levels,so that the design matrix can perform better under the criterion of uniformity measurement.As a result,good lattice point method is often used to construct uniform design.But for a long time,the uniformity of good lattice point design has not been analyzed and studied.In this paper,we will first introduce the general construction of good lattice point design,and analyze its properties under wrap-around L2-discrepancy.In addition,we also prove that the wrap-around L2-discrepancy of good lattice point design asymptotically tends to the existing lower bound of discrepancy.On this basis,we improve the conventional structure of good lattice point design,and analyze the relevant properties.It can be proved that the improved design has smaller wrap-around L1-discrepancy compared with the conventional good lattice point design.The numerical calculation also verifies our theoretical results.