Highly nonlinear functions

Let f be a function from \(\mathbb {Z}_q^m\) to \(\mathbb {Z}_q\). Such a function f is bent if all values of its Fourier transform have absolute value 1. Bent functions are known to exist for all pairs \((m,q)\) except when m is odd and \(q\equiv 2\pmod 4\) and there is overwhelming evidence that no bent function exists in the latter case. In this paper the following problem is studied: how closely can the largest absolute value of the Fourier transform of f approach 1? For \(q=2\), this problem is equivalent to the old and difficult open problem of determining the covering radius of the first order Reed–Muller code. The main result is, loosely speaking, that the largest absolute value of the Fourier transform of f can be made arbitrarily close to 1 for q large enough.