Families of Reed-Muller canonical forms

Abstract

The existence of a large number of Reed-Muller canonical forms for switching functions is demonstrated and it is shown that these can be arranged in a nested hierarchy of families with increasing size and decreasing general structure. Many of these forms can be derived, and their weights evaluated for particular functions, by employing recursively defined incidence matrices operating on an extended form of truth vector. The more useful forms are also characterized by their being realizable using a modular circuit tree involving combinations of 1-variable sub-modules.