Circulant conference matrices for new complex Hadamard matrices

TL;DR

This paper introduces a method using circulant matrices to generate new complex Hadamard matrices from conference matrices, expanding the known parametrizations especially for dimension 12.

Contribution

It presents a novel approach to constructing complex Hadamard matrices via circulant conference matrices and Sylvester inverse orthogonal matrices, providing new parametrizations for dimension 12.

Findings

New parametrizations of Hadamard matrices for n=12

Construction of complex Hadamard matrices from conference matrices

Method applicable to both real and complex matrices

Abstract

The circulant real and complex matrices are used to find new real and complex conference matrices. With them we construct Sylvester inverse orthogonal matrices by doubling the size of inverse complex conference matrices. When the free parameters take values on the unit circle the inverse orthogonal matrices transform into complex Hadamard matrices. The method is used for $n=6$ conference matrices and in this way we find new parametrisations of Hadamard matrices for dimension $ n=12$.