An extension for residue difference sets (Q1356478)

In the theory of difference sets, one of the major classes are the cyclotomic difference sets (or residue difference sets), that is, difference sets formed by (the union of) certain cyclotomic classes of residues modulo a prime \(p\) (or, more generally, of cyclotomic classes in a finite field). Unfortunately, not too many such difference sets are known to exist; in particular, no single cyclotomic class of sextic residues forms a difference set. See the monography by \textit{T. Storer} [Cyclotomy and difference sets (1967; Zbl 0157.03301)] for background. The authors consider a generalization, where the list of differences of the form \(d-md'\) is considered, where \(d\), \(d'\) run over the set \(D\) of \(n\)th power residues and where \(m\not\in D\) is called a qualifier; as usual, it is required that any non-zero residue occurs a constant number of times. (The classical case would of course be the case excluded here, namely \(m=1\) or, equivalently, \(m\in D\).) They present existence results for the quartic and the sextic case; for instance, in the sextic case, such a qualified residue difference set exists if and only if \(p\) is a prime of the form \(p= 108x^2+ 1\).