# Root-Hadamard transforms and complementary sequences

## TL;DR

This paper introduces a new root-Hadamard transform for generalized Boolean functions, unifying various existing transforms, and explores its properties and the concept of complementarity akin to Golay sequences.

## Contribution

It defines the root-Hadamard transform for generalized Boolean functions and establishes a new notion of complementarity based on this transform.

## Key findings

- The root-Hadamard transform generalizes multiple existing transforms.
- Behavior of the transform is characterized in terms of binary components.
- A new concept of complementarity for Boolean functions is introduced.

## Abstract

In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, $2^k$-Hadamard, consta-Hadamard and all $HN$-transforms. We describe the behavior of what we call the root- Hadamard transform for a generalized Boolean function $f$ in terms of the binary components of $f$. Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set.