Some Remarks on Maximally Non-Linear Functions

Summary In order to study maximally non-linear functions, we introduce a transform which is rather similar to a Fourier transform. We express the Hamming distance between a general function and a linear function using this transform. Then we study some properties of the maximal distance from a function to the generalized order one Reed-Muller code. In particular, a complete description is given for quadratics functions. By this means, we obtain a lower bound for the covering ray of the generalized order one Reed-Muller code.