Exotic complex Hadamard matrices, and their equivalence

TL;DR

This paper introduces a design theoretical approach to construct new complex Hadamard matrices, explains the existence of sporadic examples, and proposes a new invariant to distinguish inequivalent matrices, also refuting a related conjecture.

Contribution

It develops a novel design theoretical method for constructing complex Hadamard matrices and introduces the fingerprint invariant for their classification.

Findings

Constructed new complex Hadamard matrices using the proposed method.

Introduced the fingerprint invariant to distinguish inequivalent matrices.

Refuted a conjecture regarding minors of real Hadamard matrices.

Abstract

In this paper we use a design theoretical approach to construct new, previously unknown complex Hadamard matrices. Our methods generalize and extend the earlier results of de la Harpe--Jones and Munemasa--Watatani and offer a theoretical explanation for the existence of some sporadic examples of complex Hadamard matrices in the existing literature. As it is increasingly difficult to distinguish inequivalent matrices from each other, we propose a new invariant, the fingerprint of complex Hadamard matrices. As a side result, we refute a conjecture of Koukouvinos et al. on (n-8)x(n-8) minors of real Hadamard matrices.