Cyclotomy and addition sets
type: Item
instance of: scholarly article
title: Cyclotomy and addition sets
zbMATH Open document ID: 0359.05017
DOI: 10.1016/0097-3165(77)90062-0
author: Clement Lam
published in: Journal of Combinatorial Theory. Series A
publication date: 1977-01-01T00:00:00Z
review text: \(A(v,k,\lambda,g)\) - addition set \(A=(a_1,\dots,a_k)\) is a collection of \(k\) distinct residues modulo \(v\) such that any non-zero residue \(\gamma\) has exactly \(\lambda\) representations of the form \(a_i+ga_j\equiv\gamma\). If the polynomial \(\theta(x)=x^{a_1}+\dots+x^{a_k}\) is considered this leads to \(\theta(x)\theta(x^g)\equiv d+\lambda(1+x+\dots+x^{v-1})\) modulo \(x^v-1\), where \(d+\lambda\) is the number of representations of \(0\). Motivated by the application to addition sets the author studies the existence question for such polynomials \(\theta(x)\), which naturally leads to considerations of multipliers and to cyclotomy. He searches for addition sets comprised of unions of index classes for primes \(p=Nf+1\) with \(N\leq12\) and finds some for \(N=6,8,10\) and \(12\). In fact, for \(N=8\) an infinite class is constructed and for \(N=10\), \(p=41\), he finds an example with \(1<d\neq k-\lambda\) which he claims is the first such known. Using the machinery developed earlier he then examines two cases remaining from his earlier studies, (95,10,1,18) and (95,10,1,56); he shows that no such addition sets exist.
Mathematics Subject Classification ID: 05B30
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11B99
Mathematics Subject Classification ID: 11T22
zbMATH DE Number: 3559598
MaRDI profile type: Publication
full work available at URL: 0097-3165(77)90062-0
OpenAlex ID: W2060418658
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description: scientific article; zbMATH DE number 3559598