Sum sequences modulo n

A sum sequence modulo n is a sequence S = (s(1), s(2), . . . , s(d)) of elements in Z/nZ such that every x is an element of Z/nZ can be represented as s(i)+s(j), i < j, in the same number lambda of ways. For example, (0,1, 2, 4) is a sum sequence modulo 6 with lambda = 1. We examine polynomials associated with sum sequences using tools from number theory, combinatorics and Galois theory. In particular, we give a complete characterization of sum sequences and their associated polynomials. We also describe some variations on these ideas and mention several possible generalizations to arbitrary finite groups. (C) 2018 Elsevier Inc. All rights reserved.