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In Bagchi (2010) main effect plans "orthogonal through the block factor"(POTB) have been constructed. The main advantages of a POTB are that (a) it mayexist in a set up where an "usual" orthogonal main effect plan (OMEP) cannotexist and (b) the data analysis is nearly as simple as an OMEP. In the presentpaper we extend this idea and define the concept of orthogonality between apair of factorial effects ( main effects or interactions) "through the blockfactor" in the context of a symmetrical experiment. We consider plans generatedfrom an initial plan by adding runs. For such a plan we have derived necessaryand sufficient conditions for a pair of effects to be orthogonal through theblock factor in terms of the generators. We have also derived a sufficientcondition on the generators so as to turn a pair of effects aliased in theinitial plan separated in the final plan. The theory developed is illustratedwith plans for experiments with three-level factors in situations whereinteractions between three or more factors are absent. We have constructedplans with blocks of size four and fewer runs than a resolution $V$ planestimating all main effects and all but at most one two-factor interactions.

Nearly resolution V plans on blocks of small size