Complex Hadamard matrices attached to even orthogonal schemes of class 4

TL;DR

This paper constructs complex Hadamard matrices within the Bose-Mesner algebra of a specific 4-class symmetric association scheme, providing new examples and analyzing their algebraic properties.

Contribution

It introduces new constructions of complex Hadamard matrices linked to even orthogonal schemes of class 4 and investigates their algebraic decomposability.

Findings

Constructed complex Hadamard matrices in the Bose-Mesner algebra

Determined Nomura algebras showing non-decomposability

Provided explicit examples of matrices with specific algebraic properties

Abstract

A complex Hadamard matrix is a square matrix W with complex entries of absolute value 1 satisfying WW*=nI, where * stands for the Hermitian transpose and I is the identity matrix of order n. In this paper, we give constructions of complex Hadamard matrices in the Bose-Mesner algebra of a certain 4-class symmetric association scheme. Moreover, we determine the Nomura algebras to show that the resulting matrices are not decomposable into nontrivial generalized tensor products.