Abstract: The minimum aberration criterion is one of the most popular criteria for choosing optimal fractional factorial designs. This paper studies the structure of the 4^12^n designs D. For n satisfying 5N/16+1\le n\le N/2, where N is the run size of D, it first obtains the lower bounds of A_30(D) and A_31(D), the key terms that describe the optimality of D, and the structure of D which attains the lower bounds. Then it gives the construction methods of minimum aberration 4^12^n designs or weak minimum aberration 4^12^n designs.