Statistica Sinica 27 (2017), 000-000 doi:http://dx.doi.org/10.5705/ss.202015.0230 UNIFORM FOUR-LEVEL DESIGNS FROM TWO-LEVEL DESIGNS: A NEW LOOK Kashinath Chatterjee, Zujun Ou, Frederick K. H. Phoa and Hong Qin Visva-Bharati University, Jishou University, Academia Sinica and Central China Normal University Abstract: Literature reviews reveal that the research on the issue of constructing efficient uniform designs has been very active in the last decade. In addition, coding theory is widely used in the context of constructing good optimal designs. The present paper explores the construction of highly efficient four-level uniform designs via two transformations: a modified Gray map code and a mapping between quaternary codes and the sequence of three binary codes. Efficiency is based on uniformity measured by the centered L2- and wrap-around L2-discrepancies of the four-level designs’ binary images. Some results related to the lower bounds of the uniformity measures for such designs are also considered in this study. Key words and phrases: Efficiency, lower bound, modified gray map, quaternary code, uniform design. 1. Introduction Computer experiments have been widely used in engineering and high-tech- nology development because they are often cheaper and faster than physical ex- periments to perform. Unlike traditional experiments with known models, com- puter experiments are often conducted with little knowledge about the model functions. Moreover, many proposed designs of computer experiments may in- volve more than one model, which could be linear or nonlinear and parametric or nonparametric. Among many modeling methods, the uniform design performs well and becomes a central concept that plays a crucial role in the evaluation and construction of space filling designs for computer experiments (Bates et al. (1996)). In particular, in the study of model robustness, the uniform design dis- tributes its experimental points evenly throughout the design space and allows practitioners to efficiently perform numerical analyses for their experiments (see, Fang and Wang (1994, Chap. 5)). The measure of uniformity plays a key role in the construction of uniform designs. An s-level U -type design U that belongs to a design class U (n; s m ) is an n × m array with entries from the set {1/2s, 3/2s,..., (2s - 1)/2s} such that each entry of the set {1/2s, 3/2s,..., (2s -1)/2s} appears equally often in each column Statistica Sinica 27 (2017), 171-186