Spherical Designs and Generalized Sum-Free Sets in Abelian Groups

We extend the concepts of sum-freesets and Sidon-sets of combinatorial number theory with the aimto provide explicit constructions for spherical designs. We calla subset S of the (additive) abelian group Gt-free if for all non-negative integers kand l with k+l ≤ t, the sum of k(not necessarily distinct) elements of S does notequal the sum of l (not necessarily distinct) elementsof S unless k=l and the two sums containthe same terms. Here we shall give asymptotic bounds for thesize of a largest t-free set in Zn,and for t ≤ 3 discuss how t-freesets in Zn can be used to constructspherical t-designs.