We construct infinite classes of almost bent and almost perfect nonlinear polynomials, which are affinely inequivalent to any sum of a power function and an affine function.

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Following an example in [12], we show how to change one coordinate function of an almost perfect nonlinear (APN) function in order to obtain new examples. It turns out that this is a very powerful method to construct new APN functions. In particular, we show that our approach can be used to construct a "non-quadratic" APN function. This new example is in remarkable contrast to all recently constructed functions which have all been quadratic. An equivalent function has been found independently by Brinkmann and Leander [8]. However, they claimed that their function is CCZ equivalent to a quadratic one. In this paper we give several reasons why this new function is not equivalent to a quadratic one

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.

Abstract. Bent functions f : F m 2 → F2 achieve largest distance to all linear functions. Equivalently, their spectrum with respect to the Hadamard-Walsh transform is flat (i.e. all spectral values have the same absolute value). That is equivalent to saying that the function f has optimum periodic autocorrelation properties. Negaperiodic correlation properties of f are related to another unitary transform called the nega-Hadamard transform. A function is called negabent if the spectrum under the nega-Hadamard transform is flat. In this paper, we consider functions f which are simultaneously bent and negabent, i.e. which have optimum periodic and negaperiodic properties. Several constructions and classifications are presented.