Modular Golomb rulers and almost difference sets
Daniel M. Gordon
TL;DR
The paper analyzes existence questions for $(v,k,\lambda,t)$-almost difference sets and their connection to difference sets and modular Golomb rulers. It develops constructive tools for creating ADS by adding or removing a single element from a difference set, and classifies octic-residue ADS (types $O$ and $O_0$) while examining when such constructions yield valid ADS. It also studies modular Golomb rulers via relative difference sets (n=2), providing exhaustive MGR results up to $k=15$ and conjectural limits consistent with known Singer-type lifts. Complementing the theory, the author performs computational searches for general ADS with $v\le63$, corrects previous data, and maps the ADS existence landscape, including an explicit classification of residue-based ADS and open cases that guide future work.
Abstract
A $(v,k,λ)$-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 $$D D^{-1} = n + λG,$$ where $n=k-λ$. In other words, the nonzero elements of $G$ all occur exactly $λ$ times as differences of elements in $D$. A $(v,k,λ,t)$-almost difference set has $t$ nonzero elements of $G$ occurring $λ$ times, and the other $v-1-t$ occurring $λ+1$ times. When $λ=0$, 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.