On Disjoint Golomb Rulers

TL;DR

This paper investigates the structure and construction of disjoint Golomb rulers, proposing conjectures and computationally verifying values of the minimal set size needed for their existence.

Contribution

It introduces new conjectures on the construction of disjoint Golomb rulers and provides computational results for various parameters, including exact values and bounds for H(I,J).

Findings

Computed 18 exact values of H(I,J) for specified I and J.

Established 10 upper bounds on H(I,J) through computer search.

Determined H(I,J)=IJ for I > 13 and 10 ≤ J ≤ 13.

Abstract

A set $\{a_i\:|\: 1\leq i \leq k\}$ of non-negative integers is a Golomb ruler if differences $a_i-a_j$, for any $i \neq j$, are all distinct. A set of $I$ disjoint Golomb rulers (DGR) each being a $J$-subset of $\{1,2,\cdots, n\}$ is called an $(I,J,n)-DGR$. Let $H(I, J)$ be the least positive $n$ such that there is an $(I,J,n)-DGR$. In this paper, we propose a series of conjectures on the constructions and structures of DGR. The main conjecture states that if $A$ is any set of positive integers such that $|A| = H(I, J)$, then there are $I$ disjoint Golomb rulers, each being a $J$-subset of $A$, which generalizes the conjecture proposed by Koml{\'o}s, Sulyok and Szemer{\'e}di in 1975 on the special case $I = 1$. These conjectures are computationally verified for some values of $I$ and $J$ through modest computation. Eighteen exact values of $H(I,J)$ and ten upper bounds on $H(I,J)$ are obtained by computer search for $7 \leq I \leq 13$ and $10 \leq J \leq 13$. Moveover for $I > 13$ and $10 \leq J \leq 13$, $H(I,J)=IJ$ are determined without difficulty.