On Fixed Points and Cycles in the Reed Muller Domain

Abstract

This paper studies cycles that appear by repeatedly applying the RM transform to a p-valued function. It is shown that there are nontrivial fixed points, which correspond to eigenvectors of the transform and a simple method is proposed to determine the maximum period of n-place functions for a given p. The concept of spectral diversity is introduced, which may be applied to characterize pvalued functions.

On a property of the Reed-Muller-Fourier transform

“…This means that the RM spectrum or the RMF spectrum of an n-place p-valued functions is again an n-place pvalued function, not necessarily different from the original one. (It has been shown that both transforms have fixed points [8], [9]). Moreover, both the RM and the RMF transforms have a Kronecker product structure.…”

On Eigenvectors of the Pascal and Reed-Muller-Fourier Transforms

“…The eigenfunctions of the Reed-Muller transform of Boolean and multiple-valued functions were examined in [12] and [8], respectively. For the Reed-Muller-Fourier transform, the study of the eigenfunctions was initiated in [7], and the following conjecture was formulated about the number of fixed points (note that it agrees with the result of [12] for h = 2).…”