New Results on Modular Golomb Rulers, Optical Orthogonal Codes and Related Structures

TL;DR

This paper establishes new existence and nonexistence results for modular Golomb rulers and related combinatorial structures, providing a complete classification for small orders and general results for larger ones.

Contribution

It completely determines the existence of modular Golomb rulers for all orders up to 11 and introduces a general existence theorem for all larger orders, along with new nonexistence results.

Findings

Complete classification of modular Golomb rulers for k ≤ 11

General existence result for all k ≥ 3

New nonexistence results for infinite classes of related structures

Abstract

We prove new existence and nonexistence results for modular Golomb rulers in this paper. We completely determine which modular Golomb rulers of order $k$ exist, for all $k\leq 11$, and we present a general existence result that holds for all $k \geq 3$. We also derive new nonexistence results for infinite classes of modular Golomb rulers and related structures such as difference packings, optical orthogonal codes, cyclic Steiner systems and relative difference families.