We find a sufficient condition on the spectrum of a partial
geometric design d* such that, when d* satisfies this condition,
it is better (with respect to all convex decreasing optimality criteria) than
all unequally replicated designs (binary or not) with the same parameters
b, v, k as d*.
¶ Combining this with existing results, we obtain the following
results:
¶ (i) For any q \ge 3, a linked block design with parameters
b = q2, v = q2 + q,
k = q2 -1 is optimal with respect to all convex
decreasing optimality criteria in the unrestricted class of all connected
designs with the same parameters.
¶ (ii) A large class of strongly regular graph designs are optimal
w.r.t. all type 1 optimality criteria in the class of all binary designs (with
the given parameters). For instance, all connected singular group divisible
(GD) designs with \lambda_1 = \lambda_2 + 1 (with one possible exception) and
many semiregular GD designs satisfy this optimality property.
¶ Specializing these general ideas to the Acriterion, we
find a large class of linked block designs which are Aoptimal in the
unrestricted class. We find an even larger class of regular partial
geometric designs (including, for instance, the complements of a large number
of partial geometries) which are Aoptimal among all binary designs.