## Abstract

Acomplex 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 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.

Original language | English |
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Pages (from-to) | 91-110 |

Number of pages | 20 |

Journal | Special Matrices |

Volume | 3 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 Jan |

## Keywords

- Association scheme
- Complex Hadamard matrix
- Type-II matrix

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology