The Eigenfunction of the Reed-Muller Transformation

We introduce eigenfunctions of the Reed-Muller transform. Eigenfunctions are functions whose canonical sumof-products expression and PPRM (positive polarity ReedMuller expression) are isomorphic. In the case of symmetric functions, the eigenfunction can be viewed as a function whose reduced truth vector is identical to the reduced ReedMuller spectrum. We show that the number of symmetric (ordinary) eigenfunctions on -variables is ( ). We identify three special symmetric functions that correspond to the most complicated minimal fixed polarity ReedMuller (FPRM) form. We show how the transeunt triangle can be used to convert between the reduced (ordinary) truth vector and the reduced (ordinary) Reed-Muller spectrum. We derive the number of products in the FPRM for these symmetric functions: this shows that they have the most complicated minimal FPRM among all -variable functions. 1 AND-EXOR Expressions In this part, we define some classes of AND-EXOR expressions. Theorem 1.1 An arbitrary logic function can be expanded as