New lower bounds of four-level and two-level designs via two transformations

Abstract

Code theory is widely used to construct optimal designs in recent years. In this paper, two transformations, a modified Gray map and a mapping between quaternary codes and the sequence of three binary codes for four-level designs, are considered. Via the two transformations, we point out that the wrap-around $$L_2$$-discrepancy values of the two-level designs corresponding to a four-level design are decided by the four-level design, two new analytical expressions of the wrap-around $$L_2$$-discrepancy for the derived two-level designs are built, and some new lower bounds of the wrap-around $$L_2$$-discrepancy for four-level and two-level designs are obtained, which can be used as a benchmark for search the uniform designs and evaluate the uniformity of designs. Furthermore, based on the second transformation, we provide a very strong link between the aberration of a four-level design and the uniformity of the derived two-level design.