A Walsh-Fourier approach to the circulant Hadamard conjecture

TL;DR

This paper introduces a Walsh-Fourier analytical framework to study the circulant Hadamard conjecture, translating the problem into a linear algebraic system and proposing a new approach for its proof.

Contribution

It presents a novel Walsh-Fourier analysis method linking circulant Hadamard matrices to linear systems, offering a new pathway to prove the conjecture.

Findings

Equivalence between circulant Hadamard matrices and solutions to a linear system

Proposed a new approach to prove the conjecture using this system

Framework potentially simplifies the search for such matrices

Abstract

We describe an approach to the circulant Hadamard conjecture based on Walsh-Fourier analysis. We show that the existence of a circulant Hadamard matrix of order $n$ is equivalent to the existence of a non-trivial solution of a certain homogenous linear system of equations. Based on this system, a possible way of proving the conjecture is proposed.