Consider the class of balanced fractional $2^m$ factorial designs of resolution VII. Within this class, optimal designs with respect to the trace criterion are given for any fixed $N$ assemblies, which satisfy (i) $m = 6, 42 \leqq N \leqq 64$, (ii) $m = 7, 64 \leqq N \leqq 90$ and (iii) $m = 8, 93 \leqq N \leqq 128$. The covariance matrices of the estimates of the effects are also given for such designs.
"Optimal Balanced Fractional $2^m$ Factorial Designs of Resolution VII, $6 \leqq m \leqq 8$." Ann. Statist. 4 (3) 515 - 531, May, 1976. https://doi.org/10.1214/aos/1176343458