Cyclotomic orthomorphisms of finite fields
type: Item
instance of: scholarly article
title: Cyclotomic orthomorphisms of finite fields
zbMATH Open document ID: 1364.11155
author: Jordan Bell
published in: Discrete Applied Mathematics
publication date: 2012-11-22T00:00:00Z
review text: A mapping \(f\) of a group onto itself is called \textit{orthomorphism} if both \(x\mapsto f(x)\) and \(x \mapsto f(x)-x\) define bijections, and \(f\) is called a \textit{strong complete mapping} if, in addition, \(x\mapsto f(x)+x\) defines a bijection too. Such mappings have applications in cryptography, coding theory and combinatorics (for instance in the construction of designs and of mutually orthogonal sets of Latin squares). Furthermore, for primes p, strong complete mappings of \((\mathbb{F}_p,+)\) correspond to toroidal-p-queens solutions (cf.\ \textit{J. Bell} and \textit{B. Stevens} [Discrete Math. 309, No. 1, 1--31 (2009; Zbl 1228.05002)]). A useful class of orthomorphism is built by those which are constructed using cyclotomy in finite fields (see e.g. \textit{A. B. Evans} [Congr. Numerantium 101, 97--107 (1994; Zbl 0864.11064)] and \textit{H. Niederreiter} and \textit{A. Winterhof} [Discrete Math. 295, No. 1--3, 161--171 (2005; Zbl 1078.11068)]): If \(q=km+1\) is a prime power and \(\gamma\) a generator of the multiplicative group \(\mathbb{F}_q^{\ast}\), a \textit{cyclotomic mapping of index} \(k\) is a mapping \(f:\mathbb{F}_q \rightarrow \mathbb{F}_q\) defined by \(f(0)=0\) and \(f(x)=a_rx\) if \(x\in \gamma^r<\gamma^k>\) for some \(a_0, a_1, \ldots a_{k-1}\in \mathbb{F}_q\). In the present paper, the authors determine the number of cyclotomic orthomorphisms of \(\mathbb{F}_q\) of index \(3,4,5\) and \(6\) for certain classes of prime powers \(q\) and count the number of cyclotomic mappings of index \(2\) which are strong complete mappings of \(\mathbb{F}_p\) for primes \(p\).
reviewed by: Ralph-Hardo Schulz
Mathematics Subject Classification ID: 11T22
Mathematics Subject Classification ID: 12E20
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 6109951
zbMATH Keywords: orthomorphisms
zbMATH Keywords: cyclotomic mappings
zbMATH Keywords: cyclotomy
zbMATH Keywords: complete mappings
zbMATH Keywords: strong complete mappings
zbMATH Keywords: toroidal-p-queens
zbMATH Keywords: queens solutions
MaRDI profile type: Publication
full work available at URL: j.dam.2012.08.013
OpenAlex ID: W2071347182
DOI: 10.1016/J.DAM.2012.08.013
description: scientific article; zbMATH DE number 6109951