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Abstract: 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 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. [Copyright &y& Elsevier]