Balanced Optimal Saturated Main Effect Plans of the $2^n$ Factorial and Their Relation to $(v, k, \lambda)$ Configurations

Abstract

This paper characterizes balanced saturated main effect plans of the $2^n$ factorial in terms of $D'D$ rather than $X'X$, where $D$ is the $(n + 1) \times n$ treatment combination matrix and $X$ is the $(n + 1) \times (n + 1)$ design matrix. Besides this result, balanced optimal (in the sense of maximum determinant of $X'X$) saturated main effect plans of the $2^{4m-1}$ factorial are discussed for various classes of designs, each class consisting of designs having (0,0,$\cdots$ 0) and $n$ treatment combinations with exactly $t$ 1's among them. The optimality results are achieved by applying theorems associated with incidence matrices of $(\nu, k, \lambda)$ configurations. In addition results are given for designs associated with the permuted $(\nu, k, \lambda)$ configurations. Finally, the approach taken in the paper can be applied to $2^n$ factorials with $n \neq 4m - 1$.