Modular Constructions of g-Golomb Rulers

Abstract

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.