We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields by using union of cyclotomic classes of Fq of order N = 2p m 1 , where p1 is a prime and m a positive integer. Our main tools are index 2 Gauss sums, instead of cyclotomic numbers.
We prove a new characterization of weakly regular ternary bent functions via partial difference sets. Partial difference sets are combinatorial objects corresponding to strongly regular graphs. Using known families of bent functions, we obtain in this way new families of strongly regular graphs, some of which were previously unknown. One of the families includes an example in [N. Hamada, T. Helleseth, A characterization of some {3v 2 + v 3 , 3v 1 + v 2 , 3, 3}-minihypers and some [15, 4, 9; 3]-codes with B 2 = 0, J. Statist.Plann. Inference 56 (1996) 129-146], which was considered to be sporadic; using our results, this strongly regular graph is now a member of an infinite family. Moreover, this paper contains a new proof that the Coulter-Matthews and ternary quadratic bent functions are weakly regular.
We give two constructions of strongly regular Cayley graphs on finite fields F q by using union of cyclotomic classes and index 2 Gauss sums. In particular, we obtain twelve infinite families of strongly regular graphs with new parameters.
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