Disjoint difference families and their applications

TL;DR

This paper surveys the applications of difference families, introduces a unified definition for disjoint difference families, and demonstrates the equivalence of two constructions from frequency hopping sequences.

Contribution

It proposes a comprehensive definition of disjoint difference families and shows the equivalence of two existing constructions from frequency hopping sequences.

Findings

Unified definition of disjoint difference families

Two constructions from frequency hopping sequences are equivalent

Discussion on equivalence notions for sequences and families

Abstract

Difference sets and their generalisations to difference families arise from the study of designs and many other applications. Here we give a brief survey of some of these applications, noting in particular the diverse definitions of difference families and the variations in priorities in constructions. We propose a definition of disjoint difference families that encompasses these variations and allows a comparison of the similarities and disparities. We then focus on two constructions of disjoint difference families arising from frequency hopping sequences and showed that they are in fact the same. We conclude with a discussion of the notion of equivalence for frequency hopping sequences and for disjoint difference families.