Near-Optimal g-Golomb Rulers

Carlos Andres Martos Ojeda,Carlos Alberto Trujillo Solarte,David Fernando Daza Urbano

doi:10.1109/access.2021.3075877

Carlos Andres Martos Ojeda, Carlos Alberto Trujillo Solarte + Show 1 more

Open Access

https://doi.org/10.1109/access.2021.3075877

Copy DOI

Abstract

A set of positive integers A is called a $g$ -Golomb ruler if the difference between two distinct elements of A is repeated up to $g$ times. This definition is a generalization of the Golomb ruler $(g = 1)$ . In this paper, we obtain new constructions for $g$ -Golomb rulers from Golomb rulers, using these constructions we find some suboptimal 2 and 3-Golomb rulers with up to 124 marks and we prove two theorems related to extremal functions associated with this sets improving already known results.

Highlights

Intermodulation interference is the combining of several signals in a nonlinear device, producing new, unwanted frequencies
In this paper we prove the existence of an infinite number of g-Golomb rulers for a prime number of marks, we define an analog function to G(m) for g-Golomb rulers, we denoted it G(g, m) and we will find upper bound for said function
GENERALIZED GOLOMB RULERS we present the formal definition of a g-Golomb ruler, but first we will mention some concepts before

Summary

INTRODUCTION

Intermodulation interference is the combining of several signals in a nonlinear device, producing new, unwanted frequencies. In [11] Atkinson, Santoro and Urrutia estudied a generalization (when the differences are repeated g times), these are called g-Golomb rulers ( [7]), i.e, a integers set A = {a1, a2, . An}, is called a g-Golomb ruler if the non-zero difference of two elements are repeated up to g times They presented near-optimal 2-Golomb ruler with m ≤ 18 marks, but the construction was exhaustive, i.e without an algebraic method that allows to reach these rulers.

GENERALIZED GOLOMB RULERS

UPPER BOUND

CONCLUSION