Spherical designs and generalized sum-free sets in abelian groups

We extend the concepts of sum-free sets and Sidon-sets of combinatorial number theory with the aim to provide explicit constructions for spherical designs. We call a subset S of the (additive) abelian group G t-free if for all non-negative integers k and l with k+l less than or equal to t, the sum of k (not necessarily distinct) elements of S does not equal the sum of l (not necessarily distinct) elements of S unless k = l and the two sums contain the same terms. Here we shall give asymptotic bounds for the size of a largest t-free set in Z(n), and for t less than or equal to 3 discuss how t-free sets in Z(n) can be used to construct spherical t-designs.