A With-Carry Walsh Transform

Abstract

We introduce an arithmetic Walsh transform. It is a with-carry analog, based on modular arithmetic, of the usual Walsh transform of Boolean functions. This is part of our continuing effort to define and investigate with-carry analogs of discrete algebraic structures used in various aspects of communications. We develop tools for analyzing arithmetic Walsh transforms. We prove that the mapping from a Boolean function to its arithmetic Walsh transform is injective. We compute the average arithmetic Walsh transforms and the arithmetic Walsh transforms of affine functions.