Optimal Partitioned Cyclic Difference Packings for Frequency Hopping and Code Synchronization

TL;DR

This paper introduces new constructions of optimal partitioned cyclic difference packings (PCDPs) that lead to improved frequency-hopping sequences and comma-free codes, solving key existence problems for certain parameters.

Contribution

The paper presents novel methods for constructing infinite families of optimal PCDPs using almost difference sets and cyclic difference matrices, advancing the design of frequency-hopping and coding schemes.

Findings

New infinite families of optimal PCDPs are constructed.

Existence of optimal PCDPs in ${\mathbb Z}_{3m}$ is established for most m.

The constructions improve frequency-hopping and code synchronization techniques.

Abstract

Optimal partitioned cyclic difference packings (PCDPs) are shown to give rise to optimal frequency-hopping sequences and optimal comma-free codes. New constructions for PCDPs, based on almost difference sets and cyclic difference matrices, are given. These produce new infinite families of optimal PCDPs (and hence optimal frequency-hopping sequences and optimal comma-free codes). The existence problem for optimal PCDPs in ${\mathbb Z}_{3m}$, with $m$ base blocks of size three, is also solved for all $m\not\equiv 8,16\pmod{24}$.