On disjoint $(v,k,k-1)$ difference families

TL;DR

This paper surveys the history and constructions of disjoint difference families, introduces a new infinite class based on Fibonacci numbers, and establishes existence conditions in various groups.

Contribution

It clarifies past work, connects it to nearring theory, and provides new existence results for disjoint difference families in different group structures.

Findings

Revealed historical overlaps in the literature.

Derived new difference families from Fibonacci sequences.

Proved existence in all groups when prime factors meet certain conditions.

Abstract

A disjoint $(v,k,k-1)$ difference family in an additive group $G$ is a partition of $G\setminus\{0\}$ into sets of size $k$ whose lists of differences cover, altogether, every non-zero element of $G$ exactly $k-1$ times. The main purpose of this paper is to get the literature on this topic in order, since some authors seem to be unaware of each other's work. We show, for instance, that a couple of heavy constructions recently presented as new, had been given in several equivalent forms over the last forty years. We also show that they can be quickly derived from a general nearring theory result which probably passed unnoticed by design theorists and that we restate and reprove in terms of differences. We exploit this result to get an infinite class of disjoint $(v,k,k-1)$ difference families coming from the Fibonacci sequence. Finally, we will prove that if all prime factors of $v$ are congruent to 1 modulo $k$, then there exists a disjoint $(v,k,k-1)$ difference family in every group, even non-abelian, of order $v$.