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.