Highly nonlinear functions

Let f be a function from to . 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 except when m is odd and 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 , 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.