This paper presents three kinds of balanced fractional $3^m$ factorial designs 
such that the general mean and all the main effects are estimable, and furthermore (A) 
the linear by linear components of the two-factor interaction 
are estimable, and the factorial effects of the quadratic 
by quadratic and linear by quadratic ones of the two-factor 
interaction are confounded with each other, (B) the quadratic 
by quadratic ones of the two-factor interaction are estimable, and the effects 
of the linear by linear and linear by quadratic ones of the two-factor interaction 
are confounded with each other, and (C) the linear by quadratic ones 
of the two-factor interaction are estimable, and the effects of the linear 
by linear and quadratic by quadratic ones of the two-factor interaction are confounded 
with each other, where the three-factor and higher-order interactions are assumed 
to be negligible and the number of assemblies is less than the number 
of non-negligible factorial effects. These designs are concretely given 
by the indices of a balanced array of full strength, which is called a simple array.