Modular Golomb Rulers and Almost Difference Sets

A $(v,k,\lambda)$ -difference set in a group G of order v is a subset $\{d_{1}, d_{2}, \ldots ,d_{k}\}$ of G such that $D=\sum d_{i}$ in the group ring ${\mathbb {Z}}[G]$ satisfies: $D D^{-1} = n + \lambda G$ , where $n=k-\lambda $ . In other words, the nonzero elements of G all occur exactly $\lambda $ times as differences of elements in D. A $(v,k,\lambda ,t)$ -almost difference set has t nonzero elements of G occurring $\lambda $ times, and the other $v-1-t$ occurring $\lambda +1$ times. When $\lambda =0$ , this is equivalent to a modular Golomb ruler. In this paper we investigate existence questions on these objects, and extend previous results constructing almost difference sets by adding or removing an element from a difference set. We also show for which primes the octic residues, with or without zero, form an almost difference set.