For the $2^n$ factorial a treatment design to estimate the $n$ main effects and the mean with $(n + 1)$ treatment combinations is known in the literature as a saturated main effect plan. Let the $(n \times 1) \times n$ matrix $D$ consisting of the 0's and 1's making up the subscripts of the observations, denote such a plan and let the $(n + 1) \times (n + 1)$ matrix $X$ stand for the corresponding design matrix of -1's and 1's, then optimal (in the sense of maximum absolute value of the determinant of $X'X$) designs have been characterized in terms of the information matrix $X'X$ by many authors, such as Plackett and Burman  and Raghavarao . Williamson , Mood , and Banerjee , among others, have used (0, 1)-matrices to construct optimal and weighing designs. If the elements of the first row of $D$ are set equal to zero, then the $n \times n$ (0, 1)-matrix used in weighing designs is obtained from the last rows of $D$. However, $D$ is not restricted to always include the combination having all zero levels in this paper. For a summary concerning several aspects of optimal saturated main effect plans the reader is referred to Addelman's  paper. The aim of this paper is to characterize the optimal saturated main effect plans in terms of D'D rather than $X'X$. A major consequence of this is that all theory available for semi-normalized $(-1, 1)$-matrices is applicable to semi-normalized (0, 1)-matrices and vice versa. A second major consequence is that the normal equations for saturated main effect plans need not be obtained as they are readily derivable from the $D$ matrix.
B. L. Raktoe. W. T. Federer. "Characterization of Optimal Saturated Main Effect Plans of the $2^n$ Factorial." Ann. Math. Statist. 41 (1) 203 - 206, February, 1970. https://doi.org/10.1214/aoms/1177697201