Sets in Z_n with Distinct Sums of Pairs

A subset S = {s_1, ..., s_k} of an abelian group G is called an S_t-set of size k if all sums of t different elements in S are distinct. A function with applications in coding theory, v_γ(k) denotes the order of the smallest cyclic group in which an S_2-set of size k exists. A lower bound for v_γ(k) is given in this study, and exact values of v_γ(k) are obtained for k ≤ 15. For the related problem in which all sums of any two, not necessarily distinct, elements in S are required to be different, values of the corresponding function v_γ(k) for each k ≤ 14 are given