Nonlinear functions in abelian groups and relative difference sets.
type: Item
instance of: scholarly article
title: Nonlinear functions in abelian groups and relative difference sets.
zbMATH Open document ID: 1035.05023
DOI: 10.1016/S0166-218X(03)00293-2
author: Alexander Pott
published in: Discrete Applied Mathematics
publication date: 2004-03-29T00:00:00Z
review text: This paper shows that the main results on nonlinear functions on finite fields can be generalized to abelian groups using the discrete Fourier tranform. The paper is a very interesting survey on (generalisations of) relative difference sets and nonlinear functions introducing new points of view. Let \(K\) and \(N\) be abelian (additive) groups with \(| K| =m\), \(| N| =n\) and \(f:K\to N\) be a function. Let \(D_f=\{\langle g,f(g)\rangle \mid g\in K\}\subset G=K\times N\). Call a function \(f\) perfect nonlinear if \(\delta_f(a,b)=| \{g\in K\mid f(g+a)-f(g)=b\}| \) is equal to \(m/n\) for all \(a\in K\setminus\{0\}\) and \(b\in N\). Now \(D_f\) is a splitting \((m,n,m,m/n)\)-DS in \(G\) relative to \(\{0\}\times N\) if and only if \(f\) is perfect nonlinear. This suggest to use the discrete Fourier transform to study more general sets \(D_f\). A key of the paper is the following definition: a function \(f:K\to N\) is an almost perfect nonlinear function if \(\sum_{a,b}[\delta_f(a,b)]^2\leq\sum_{a,b}[\delta_g(a,b)]^2\) for all functions \(g:K\to N\). Too many definitions and notations are necessary to go here into detail; we can only note that using this new point of view many proofs are more transparent and connections with relative difference sets become apparent.
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 20K99
zbMATH DE Number: 2062822
zbMATH Keywords: almost perfect nonlinear functions
reviewed by: Giovanni Ferrero
MaRDI profile type: Publication
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full work available at URL: s0166-218x(03)00293-2
OpenAlex ID: W2154374104
description: scientific article; zbMATH DE number 2062822