Modular Constructions of g-Golomb Rulers

A set $\mathcal{G}$ of integers is a $g$-Golomb ruler if each positive difference appears at most $g$ times between any 2 elements of the set, and $G(g,n)$ denotes the minimum diameter of such a ruler with $n$ marks. We prove a general lemma for passing from certain modular constructions to ordinary $g$-Golomb rulers. The key point is that, in a modular $g$-Golomb ruler, no cyclic gap length can occur more than $g$ times. This gives a larger guaranteed cut than the previous average gap argument. We apply this lemma to cyclic relative difference sets, Singer sets, Ruzsa–Spence rulers, and Paley quadratic residues to provide many competing constructions for $g$-Golomb Rulers. A computation on the grid $1\le g\le 500$, $n=g+b$, $2\le b\le 500$, compares the four resulting construction families.