This paper presents the trace of the covariance matrix of the estimates of effects based on a fractional 2^m factorial (2^m-FF) design T of resolution V for the following two cases: One is the case where T is constructed by adding some restricted assemblies to an orthogonal array. The other is one where T is constructed by removing some restricted assemblies from an orthogonal array of index unity. In the class of 2^m-FF designs of resolution V considered here, optimal designs with respect to the trace criterion, i.e. A-optimal, are presented for m = 4, 5, and 6 and for a range of practical values of N (the total number of assemblies). Some of them are better than the corresponding A-optimal designs in the class of balanced fractional 2^m factorial designs of resolution V obtained by Srivastava and Chopra (1971b) in such a sense that the trace of the covariance matrix of the estimates is small.