On the Reed-Muller spectrum of symmetric boolean functions

Abstract

The paper considers a class of symmetric Boolean functions called Reed-Muller type Fundamental Symmetric Functions, it reviews some of their properties and presents some new ones. The main contribution of the paper is a proof that the Reed-Muller transform of a symmetric Boolean function is also symmetric and that of a rotation symmetric Boolean function is also rotation symmetric. Since symmetric n-place Boolean functions may be given a compact representation with a value vector of n+1 elements and this holds also for its Reed-Muller spectrum, some methods are reviewed, to calculate the Reed-Muller spectrum of a symmetric Boolean function based on its compact value vector. Furthermore a method is presented to calculate the Reed-Muller spectrum of a rotation symmetric Boolean function from the compact value vector representation of the function.