Uniformity of Mixed Two- and Four-Level Designs under Quaternary Codes

In this paper, mixed two- and four-level designs are transformed to two-level designs via two types of quaternary coding mapping, and the uniformity measured by wrap-around $L_2$-discrepancy of mixed two- and four-level designs is studied, respectively. Some lower bounds of wrap-around $L_2$-discrepancy for mixed two- and four-level designs are obtained, three numerical examples are provided to illustrate the theoretical results, and it is shown that these lower bounds are tight, which can serve as a benchmark in the construction of uniform mixed two- and four-level designs.