  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 may
exist in a set up where an "usual" orthogonal main effect plan (OMEP) cannot
exist and (b) the data analysis is nearly as simple as an OMEP. In the present
paper we extend this idea and define the concept of orthogonality between a
pair of factorial effects ( main effects or interactions) "through the block
factor" in the context of a symmetrical experiment. We consider plans generated
from an initial plan by adding runs. For such a plan we have derived necessary
and sufficient conditions for a pair of effects to be orthogonal through the
block factor in terms of the generators. We have also derived a sufficient
condition on the generators so as to turn a pair of effects aliased in the
initial plan separated in the final plan. The theory developed is illustrated
with plans for experiments with three-level factors in situations where
interactions between three or more factors are absent. We have constructed
plans with blocks of size four and fewer runs than a resolution $V$ plan
estimating all main effects and all but at most one two-factor interactions.
