Complex Hadamard matrices contained in a Bose-Mesner algebra

TL;DR

This paper constructs complex Hadamard matrices within Bose-Mesner algebras of certain association schemes, providing new examples, analyzing their properties, and confirming their inequivalence and indecomposability.

Contribution

It introduces a method to construct complex Hadamard matrices in Bose-Mesner algebras using rational maps, including new matrices of order 15, and analyzes their algebraic properties.

Findings

Constructed new complex Hadamard matrices in Bose-Mesner algebras

Recovered known order 15 Hadamard matrices by Ada Chan

Proved the matrices are inequivalent and indecomposable

Abstract

A complex Hadamard matrix is a square matrix H with complex entries of absolute value 1 satisfying $HH^*= nI$, where $*$ stands for the Hermitian transpose and I is the identity matrix of order $n$. In this paper, we first determine the image of a certain rational map from the $d$-dimensional complex projective space to $\mathbb{C}^{d(d+1)/2}$. Applying this result with $d=3$, we give constructions of complex Hadamard matrices, and more generally, type-II matrices, in the Bose-Mesner algebra of a certain 3-class symmetric association scheme. In particular, we recover the complex Hadamard matrices of order 15 found by Ada Chan. We compute the Haagerup sets to show inequivalence of resulting type-II matrices, and determine the Nomura algebras to show that the resulting matrices are not decomposable into generalized tensor products.