Generalized Bent Functions and Their Properties

摘要

Let J(m/q) denote the set of m-tuples over the integers modulo q and set i = sq. rt. (-1), w = exp(i(2pi/q)). As an extension of Rothaus' notion of a bent function, a function, f: J(m/q) yields J(1/q) is called bent if all the Fourier coefficients of w superscript f have unit magnitude. An important feature of these functions is that their out of phase autocorrelation value is identically zero. The nature of the Fourier coefficients of a bent function is examined and proof for the non-existence of bent functions over J(m/q), m odd, is given for many values of q of the form q=2(mod 4). For every possible value of q and m (other than m odd and q=2(mod 4)), constructions of bent functions are provided. Keywords: Generalized bent functions; Hadamard difference sets; Automorphism; Complex conjugation; Cyclotomic fields; Fourier coefficients; Hadamard matrices; Coset; Prime; Subgroup; Uniqueness; and Reprints.