Three-level designs for definitive screening, originally introduced as definitive screening designs (DSDs) by Jones and Nachtsheim (J Qual Technol 43(1): 1–15, 2011, https://doi.org/10.1080/00224065.2011.11917841), have been extensively studied during the last decade because of their attractive properties for factor screening. These designs have columns with three levels and possess a fold-over structure which guarantees nice properties under a full second order linear model. The original DSDs have one centre point, a total of three zeros per column, and study q factors in \(n=2q+1\) runs. In recent years, three-level designs that possess the form and the attractive properties of the original DSDs have been proposed, where the number of zeros per column is greater than three and the number of factors under study is less than q. In this paper, we study the correlation between the columns that correspond to pure quadratic effects in three-level designs for definitive screening with n runs, s centre points and z zeros per column. Then, we construct all non-isomorphic designs with n runs and three columns with up to seven zeros per column that attain the minimum possible correlation among pure quadratic columns, for every run order n. Finally, for specific run orders \(n \le 33\) , we exploit the lists of non-isomorphic \(n \times 3\) designs in an extension algorithm, to construct designs with \(n \le 33\) runs and more than three columns with three, five and seven zeros per column, that have the minimum possible correlation among quadratic columns. Our method can be easily modified to cover design construction cases with more than seven zeros per column.
