Strongly regular graphs associated with ternary bent functions

Abstract

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 { 3 v 2 + v 3 , 3 v 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.