A catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs

The foldover plan is a transformation map for adding a foldover design to the initial design, thus resulting in a combined design which can be used for breaking the links between aliased effects. This paper discusses the optimality of foldover plans for four-level designs via the most common criteria: the generalized word-length pattern (GWLP), Lee discrepancy (LD), wrap-around discrepancy (WD), centered discrepancy (CD) and mixture discrepancy (MD). We prove that the LD, WD and GWLP are equivalent for: any initial design and the corresponding foldover designs; any foldover design and its complementary foldover design; and any combined design and its complementary combined design. However, these interesting properties do not necessarily take place in the cases of the CD and MD. New analytical expressions and lower bounds of these discrepancies are given for initial and combined designs, which can be used as benchmarks for constructing uniform designs. For illustration of the usage of our theoretical results, a catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs that involve $ 2\leq m\leq 10 $ 2≤m≤10 factors with $ 8\leq n\leq 52 $ 8≤n≤52 run is tabulated, which can be used for investigating either qualitative or quantitative factors.