Convolution on finite groups and fixed-polarity polynomial expressions

Abstract

This paper discusses relationships among convolution matrices and fixed-polarity matrices for polynomial expressions of discrete functions on finite groups. Switching and multiple-valued functions are considered as particular examples of discrete functions on finite groups. It is shown that if the negative literals for variables are defined in terms of the shift operators on domain groups, then there is a relationship between the polarity matrices and convolution matrices. Therefore, the recursive structure of polarity matrices follows from the recursive structure of convolution matrices. This structure is determined by the assumed decomposition of the domain groups for the considered functions.