New cyclic difference sets with Singer parameters constructed from \(d\)-homogeneous functions
type: Item
instance of: scholarly article
title: New cyclic difference sets with Singer parameters constructed from \(d\)-homogeneous functions
zbMATH Open document ID: 1059.05021
DOI: 10.1023/B:DESI.0000036246.52472.81
author: Jong-Seon No
published in: Designs, Codes and Cryptography
publication date: 2005-01-17T00:00:00Z
review text: Cyclic difference sets with the classical (Singer) parameters \((\frac{q^n-1}{q-1}, q^{n-1}, q^{n-1}-q^{n-2})\) are of interest in design theory (since we can construct designs from it) and because of the relation to sequences with good autocorrelation properties. In the case \(q=2\), many new examples of difference sets have been constructed recently, see, for instance [\textit{J. F. Dillon} and \textit{H. Dobbertin}, Finite Fields Appl. 10, 342--389 (2004; Zbl 1043.05024)]. Since then, some work has been done to generalize these binary constructions to the nonbinary case \(q>2\). In this paper, the author suggests to use \(d\)-homogeneous functions (\(f:{\mathbb F}_{q^n}\to{\mathbb F}_{q^n}, f(yx)=y^df(x)\) for \(y\in{\mathbb F}_q\)) which are also difference-balanced (the multiset \((f(yx)-f(x):x\in{\mathbb F}_{q^n})\) contains every element in \({\mathbb F}_q\) exactly \(q^{n-1}\) times). The main theorem states that difference-balanced \(d\)-homogeneous functions yield difference sets with the classical parameters. It turns out that many of the known constructions of difference sets can be represented in this way. The paper contains a new series of examples where \(q=3^k\), generalizing the construction in [\textit{T. Helleseth, P. V. Kumar} and \textit{H. Martinsen}, Des. Codes Cryptography 23, No. 2, 157--166 (2001; Zbl 1007.94013)].
reviewed by: Alexander Pott
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 2128558
zbMATH Keywords: difference set
zbMATH Keywords: autocorrelation
zbMATH Keywords: sequence
zbMATH Keywords: finite field
zbMATH Keywords: designs
MaRDI profile type: Publication
Recommended article: Difference balanced functions and their generalized difference sets
Recommended article: Some Results on Difference Balanced Functions
Recommended article: Q3641073
Recommended article: Cyclic \((v;r,s;\lambda)\) difference families with two base blocks and \(v \leq 50\)
Recommended article: Disjoint difference families and their applications
Recommended article: Difference systems of sets and cyclotomy
Recommended article: Q4026149
Recommended article: Some Notes on d-Form Functions with Difference-Balanced Property
Recommended article: Q4848761
Recommended article: Optimal and perfect difference systems of sets from \(q\)-ary sequences with difference-balanced property
description: scientific article; zbMATH DE number 2128558