Optimality of partial geometric designs

We find a sufficient condition on the spectrum of a partial geometric desig n d* such that, when d* satisfies this condition, it is better (with respec t to all convex decreasing optimality criteria) than all unequally replicat ed designs (binary or not) with the same parameters b, v, h as d*. Combining this with existing results, we obtain the following results: (i) For any q greater than or equal to 3, a linked block design with parame ters b = q(2), = q(2)+q, k = q(2)-1 is optimal with respect to all convex d ecreasing optimality criteria in the unrestricted class of all connected de signs with the same parameters. (ii) A large class of strongly regular graph designs are optimal w.r.t. all type I optimality criteria in the class of all binary designs (with the gi ven 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 A-criterion, we find a large class of linked block designs which are A-optimal in the un-restricted class. We find an even larger class of regular partial geometric designs (including, for instance, the complements of a large number of partial geometries) whic h are A-optimal among all binary designs.