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 $1\le g\le500$, $n=g+b$, $2\le b\le500$, compares the four resulting construction families.