On Constructing the Fractional Replicates of the $2^m$ Designs with Blocks
The problem of constructing fractional replicates of the $s^m$ designs, where $s$ is a prime power is not new in literature. There are several papers which deal with this problem. However, so far as the subject matter of this paper is concerned, the contributions made by Banerjee , Rao , Dykstra  and very recently by Addelman  are of special interest. This is yet another attempt in the same direction. The basic concept is the same as in , where Rao gives a method of obtaining block designs for the fractional replicates of the $s^m$ designs so as to estimate the main effects and the two-factor interactions orthogonally assuming all other interactions to be absent. With the same assumptions, a method of construction is given in this paper which gives in many cases block designs for the fractional replicates of the $2^m$ designs with lesser number of treatment combinations than that of the corresponding fractional designs given earlier. This is achieved by allowing the estimates to be correlated. The scheme allows the estimates of treatment-effects and block effects to be mutually orthogonal. An additional example is given at the end to indicate the possibility of improving the construction
Patel, M. S. (1962). On Constructing the Fractional Replicates of the $2^m$ Designs with Blocks. Annals of Mathematical Statistics, 33(4), 1440-1449.