On subsets of GF\((q^2)\) with \(d\)th power differences
type: Item
instance of: scholarly article
title: On subsets of GF\((q^2)\) with \(d\)th power differences
zbMATH Open document ID: 0945.51004
DOI: 10.1016/S0012-365X(99)00096-5
published in: Discrete Mathematics
publication date: 2000-10-08T00:00:00Z
review text: Let \(X\) be a subset with \(q\) elements of the Galois field \(GF (q^2)\) such that \(0,1\in X\) and \(x-y\) is a square for all \(x,y\in X\). In [\textit{A. Blokhuis}, Indagationes Math. 46, 369-372 (1984; Zbl 0561.12009)] it has been proved that for \(q\) odd such a subset \(X\) is the subfield \(GF(q)\). This result was conjectured in [\textit{J. H. Van Lint} and \textit{F. J. Mac Williams}, IEEE Trans. Inf. Theory IT-24, 730-737 (1978; Zbl 0395.94025)]. In this paper the author generalizes Blokhuis's result for \(d\)th powers with \(d\mid (q+1)\). Precisely he proves that a subset \(S\) of \(GF(q^2)\), with the properties \(0,1\in S\) and \(x-y\) is a \(d\)th power for all \(x,y\in S\), is the subfield \(GF (q)\). Also, he gives the following embedding condition for the set \(S\): if \(|S|>q- (1-1/d) \sqrt q\) (with the same properties as above) then \(S\subseteq GF(q)\).
reviewed by: Guglielmo Lunardon
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 94B15
zbMATH DE Number: 1369808
zbMATH Keywords: \(d\)-th power difference
zbMATH Keywords: Galois field \(GF(q^2)\)
zbMATH Keywords: embedding condition
author: Peter Sziklai
MaRDI profile type: Publication
Wikidata QID: Q127212402
description: scientific article; zbMATH DE number 1369808