A theoretical question on fractional factorials Very much in need of help interpreting this if anyone is available.. Thank you!

Fractional factorials that include many factors can sometimes have a surprisingly low number of experimental conditions that need to be run. A design with 15 factors (all of them two levels) in a full factorial would need 32,768 factorial combinations, requiring tens of thousand of subjects. However one may run a 1/2048 fraction of the design with just 16 conditions! This efficiency gain, however comes with a trade-off. What additional assumptions are necessary to interpret the result of fractional designs? Note: you do not have to derive the aliasing structure for this specific example, but simply comment in general about the trade-off in fractional factorials.
 A: You can read off the additional assumptions from the alias structure. For your example a $\mathbf{2}^{15}$ design needs, as you said, $32768$ experimental runs, but with that you can estimate even the 15-factor full interaction. Such many-factor interactions are seldom interpretable, and if you restrict yourself to main effects and two-factor interactions, the total number of parameters to be estimated is $1+15+\binom{15}{2}=1+15+105=121$ parameters, which can be accommodated with a $\mathbf{2}^{15-8}$ design.
But there are many ways to choose such a design, you need the concept of resolution of the design, see Intuition to the Resolution of a fractional factorial design. You can find a resolution V $\mathbf{2}^{15-8}$-design, if that is too many observations, and you can accept to alias some two-factor interactions, you can find a smaller resolution IV design. What you need to decide is which main affects and interactions you really need unaliased estimates for, and which you can ignore.  It could also be wise to replicate the design twice (in two blocks) to admit a pure error variance estimate. See the linked post for references.
