A two-parameter family of complex Hadamard matrices of order 6 induced by hypocycloids

TL;DR

This paper introduces a new two-parameter family of complex Hadamard matrices of order 6, extending known families and demonstrating the existence of a related 2-parameter family of mutually unbiased bases triplets.

Contribution

It presents a novel block-construction method for order 6 Hadamard matrices, expanding the known families and linking to mutually unbiased bases.

Findings

Discovered a new 2-parameter family of complex Hadamard matrices of order 6.

Extended the existing families D_6(t) and B_6(t).

Proved the existence of a 2-parameter family of MUB-triplets of order 6.

Abstract

Constructions of Hadamard matrices from smaller blocks is a well-known technique in the theory of real Hadamard matrices: tensoring Hadamard matrices and the classical arrays of Williamson, Ito are all procedures involving smaller order building blocks. We apply a new block-construction for order 6 to obtain a previously unknown 2-dimensional family of complex Hadamard matrices. Our results extend the families D_6(t) and B_6(t) found by various authors recently. As a direct application the existence of a 2-parameter family of MUB-triplets of order 6 is shown.