Modular Golomb rulers and almost difference sets

A $(v,k,\lambda )$-difference set in a group $G$ of order $v$ is a subset $\{d_1,d_2,\dots ,d_k\}$ 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.