Abstract: There exist nonregular two-level designs with run sizes a power of 2. Many of these designs have a defining relation. This article characterizes nonregular two-level fractional factorial designs according to three types. First, there are designs that can be constructed using generators that are linear combinations of orthogonal interactions from a subset of the factors. All possible generators for 16-and 32-run designs are identified. A second type of orthogonal two-level designs has partial replication, which requires adding one or more dummy factors to obtain generators. Intermediate to these two types are orthogonal designs that have no partial replication, but require augmentation in order to obtain generators. This classification and subsequent insight benefit the construction and characterization of nonregular designs. Designs of the first type have a defining relation that is easily produced from the generators. For the other cases, generators are useful for obtaining the indicator function or the extended word length pattern. Given familiarity with regular two-level fractional factorial designs, this article can serve as a bridge to understanding nonregular fractions.

Key words and phrases: Defining relation, extended word length pattern, fractional factorial, indicator function, orthogonal array, partial replication.