In this paper, based on the assumption that the four-factor and higher-order interactions are to be negligible, we consider a balanced fractional 2 ^m factorial design derived from a simple array such that all the main effects are estimable, i.e., resolution R^*({1}|3). In this situation, using the algebraic structure of the triangular multidimensional partially balanced association scheme and a matrix equation, we can get designs of four types of resolutions: the first is of resolution R({1}|3), the second is of resolution R({0,1}|3), the third is of resolution R({1,2}|3), i.e., resolution VI, and the last is of resolution R({0,1,2}|3), i.e., resolution VI. This paper gives the characterization of designs mentioned above, and also it gives optimal designs with respect to the generalized A-optimality criterion for 6 ≤ m ≤ 8 when the number of assemblies is less than the number of non-negligible factorial effects.