On subsets of GF\((q^2)\) with \(d\)th power differences
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 \(S\subseteq GF(q)\).
scientific article; zbMATH DE number 1369808