Cyclic difference sets with parameters \((511,255,127)\) (Q1332389)

We describe the (equivalence classes of) cyclic difference sets with parameters (511, 255, 127). There are five non-isomorphic such classes which are listed at the end of this note. We give also a version of their triple intersection polynomial (our version differs by a numerical factor from the polynomial used in [\((*)\) \textit{S. Eliahou} and \textit{M. Kervaire}, Enseign. Math., II. Sér. 38, No. 3/4, 345-382 (1992; Zbl 0777.05024)]). The first and second belong to known families and are obtained by a geometric construction and by the GMW-construction described (for example) in \((*)\). The last three appear to be exotic.