Nonregular two-level designs of resolution IV or more containing clear two-factor interactions

In this paper, the concepts of clear effects, alias sets and grid representations are generalized to nonregular two-level designs. Many good generalized join designs of n   runs with resolution IV or more containing many clear two-factor interactions are given for n=48n=48 up to 192 and n being a multiple of 16. The designs constructed are shown to have concise generalized two-factor interaction grid representations. Finally, theoretical results for the necessary and sufficient conditions under which there exist nonregular two-level designs of resolution IV or more containing clear two-factor interactions are proved.