An extension for residue difference sets
type: Item
instance of: scholarly article
title: An extension for residue difference sets
zbMATH Open document ID: 0884.05020
DOI: 10.1016/S0012-365X(96)00243-9
author: Derek Jennings
author: Kevin Byard
published in: Discrete Mathematics
publication date: 1998-04-01T00:00:00Z
review text: 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 \(p= 108x^2+ 1\).
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 11A15
zbMATH DE Number: 1018529
zbMATH Keywords: cyclotomic difference sets
zbMATH Keywords: residue difference sets
zbMATH Keywords: qualified residue difference set
reviewed by: Dieter Jungnickel
MaRDI profile type: Publication
cites work: Cyclic difference sets
cites work: Q5842449
cites work: Cyclotomy, Higher Congruences, and Waring's Problem
cites work: Q5519692
cites work: On Residue Difference Sets
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description: scientific article; zbMATH DE number 1018529