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We consider a fractional 2 m factorial design derived from a simple array (SA) such that the ( + 1)-factor and higher-order interactions are negligible, where 2 ≤ m. The purpose of this article is to give a necessary and sufficient condition for an SA to be a balanced fractional 2 m factorial design of resolution 2 + 1. Such a design is concretely characterized by the suffixes of the indices of an SA.

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We consider a fractional 3 m factorial design for the second-order model derived from a simple array (SA), where m ≥ 4. In this article, we give a necessary and sufficient condition for an SA to be a balanced second-order design for 3 m factorials. Such a design is characterized by the suffixes of the index of an SA.