Spectra of a class of quadratic functions: average behaviour and counting functions

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Official URL: http://dx.doi.org/10.1007/s12095-015-0142-9

The Walsh transform (Q) over cap of a quadratic function Q : F-p(n) -> F-p satisfies vertical bar(Q) over cap vertical bar is an element of {0, p(n+s/2)} for an integer 0 <= s <= n - 1. We study quadratic functions given in trace form Q( x) = Tr-n(Sigma(k)(i=0) a(i)x(pi+1)) with the restriction that a(i) is an element of F-p, 0 <= i <= k. We determine the expected value for the parameter s for such quadratic functions from F-p(n) to F-p, for many classes of integers n. Our exact formulas confirm that on average the value of s is small, and hence the average nonlinearity of this class of quadratic functions is high when p = 2. We heavily use methods, recently developed by Meidl, Topuzo. glu and Meidl, Roy, Topuzo. glu in order to construct/enumerate such functions with prescribed s. In the first part of this paper we describe these methods in detail and summarize the counting results.