Optimality of partial geometric designs

Abstract

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.