Binary (k, k)-Designs

We introduce and investigate binary (k,k)-designs, a special case of T-designs. Our combinatorial interpretation relates (k,k)-designs to the binary orthogonal arrays. We derive a general linear programming bound and propose as a consequence a universal bound on the minimum possible cardinality of (k,k)-designs for fixed k and n. Designs which attain our bound are investigated.

On Euclidean Designs and Potential Energy

We study Euclidean designs from the viewpoint of the potential energy. For a finite set in Euclidean space, we formulate a linear programming bound for the potential energy by applying harmonic analysis on a sphere. We also introduce the concept of strong Euclidean designs from the viewpoint of the linear programming bound, and we give a Fisher type inequality for strong Euclidean designs. A finite set on Euclidean space is called a Euclidean $a$-code if any distinct two points in the set are separated at least by $a$. As a corollary of the linear programming bound, we give a method to determine an upper bound on the cardinalities of Euclidean $a$-codes on concentric spheres of given radii. Similarly we also give a method to determine a lower bound on the cardinalities of Euclidean $t$-designs as an analogue of the linear programming bound.