Some Results on $2^{n - k}$ Fractional Factorial Designs and Search for Minimum Aberration Designs

In this paper we find several interesting properties of $2^{n-k}$ fractional factorial designs. An upper bound is given for the length of the longest word in the defining contrasts subgroup. We obtain minimum aberration $2^{n-k}$ designs for $k = 5$ and any $n$. Furthermore, we give a method to test the equivalence of fractional factorial designs and prove that minimum aberration $2^{n - k}$ designs for $k \leq 4$ are unique.