### Abstract

We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-Type complex Hadamard matrix whose entries are 4-Th roots of unity.We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of X. It is shown that such a matrix is again necessarily a Butson-Type complex Hadamard matrix whose entries are 4-Th roots of unity.

Original language | English |
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Pages (from-to) | 1-10 |

Number of pages | 10 |

Journal | Special Matrices |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

### Keywords

- Galois ring
- association scheme
- complex Hadamard matrix

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology