Some Results on $s^{n-k}$ Fractional Factorial Designs with Minimum Aberration or Optimal Moments

The minimum aberration criterion is commonly used for selecting good fractional factorial designs. In this paper we obtain minimum aberration $2^{n - k}$ designs for $k = 3, 4$ and any $n$. For $k > 4$ analogous results are not available for general $n$ since the resolution criterion is not periodic for general $n$ and $k > 4$. However, it can be shown that for any fixed $k$, both the resolution criterion and the minimum aberration criterion have a periodicity property in $n$ for $s^{n - k}$ designs with large $n$. Furthermore, the optimal-moments criterion is periodic for any $n$ and $k$.