A transformation on boolean functions, its consequences on some problems related to reed-muller codes

Abstract

We introduce a transformation defined on the set of all boolean functions defined on Galois fields GF(2^m), m∈ℕ*,(or on F ^m₂ ), which changes their weights in a way easy to be followed, and which, when we iterate it, reduces their degrees down to 2 or 3. We deduce that it is as difficult to find a general characterization of the weights in the Reed-Muller codes of order 3 as it is to obtain one in the Reed-Muller codes of any orders. We also use this transformation to characterize the existence of some affine sets of bent functions, and to obtain bent functions of degree 4 from bent functions of degree 3.