Enumeration of Quadratic Functions With Prescribed Walsh Spectrum

The Walsh transform (f) over cap of a quadratic function f : F-p(n) -> F-p satisfies vertical bar(f) over cap vertical bar epsilon{0, p(n+s/2)} for an integer 0 <= s <= n-1, depending on f. In this paper, quadratic functions of the form F-p,F-n(x) = Tr-n(Sigma(k)(i=0) a(i)x(pt+1)) are studied, with the restriction that a(i) is an element of F-p, 0 <= i <= k. Three methods for enumeration of such functions are presented when the value for s is prescribed. This paper extends earlier enumeration results significantly, for instance, the generating function for the counting function is obtained, when n is odd and relatively prime to p, or when n = 2m, for odd m and p = 2. The number of bent and semibent functions for various classes of n is also obtained.