New cyclic difference sets with Singer parameters constructed from \(d\)-homogeneous functions
Cyclic difference sets with the classical (Singer) parameters \((\frac{q^n-1}{q-1}, q^{n-1}, q^{n-1}-q^{n-2})\) are of interest in design theory (since we can construct designs from it) and because of the relation to sequences with good autocorrelation properties. In the case \(q=2\), many new examples of difference sets have been constructed recently, see, for instance [\textit{J. F. Dillon} and \textit{H. Dobbertin}, Finite Fields Appl. 10, 342--389 (2004; Zbl 1043.05024)]. Since then, some work has been done to generalize these binary constructions to the nonbinary case \(q>2\). In this paper, the author suggests to use \(d\)-homogeneous functions (\(f:{\mathbb F}_{q^n}\to{\mathbb F}_{q^n}, f(yx)=y^df(x)\) for \(y\in{\mathbb F}_q\)) which are also difference-balanced (the multiset \((f(yx)-f(x):x\in{\mathbb F}_{q^n})\) contains every element in \({\mathbb F}_q\) exactly \(q^{n-1}\) times). The main theorem states that difference-balanced \(d\)-homogeneous functions yield difference sets with the classical parameters. It turns out that many of the known constructions of difference sets can be represented in this way. The paper contains a new series of examples where \(q=3^k\), generalizing the construction in [\textit{T. Helleseth, P. V. Kumar} and \textit{H. Martinsen}, Des. Codes Cryptography 23, No. 2, 157--166 (2001; Zbl 1007.94013)].
scientific article; zbMATH DE number 2128558
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