Butson-type complex Hadamard matrices and association schemes on Galois rings of characteristic 4

TL;DR

This paper characterizes nonsymmetric hermitian complex Hadamard matrices within association schemes on Galois rings of characteristic 4, showing they are necessarily of Butson type with 4-th roots of unity.

Contribution

It provides a classification of hermitian complex Hadamard matrices in association schemes on Galois rings of characteristic 4, revealing their necessity to be of Butson type.

Findings

Hermitian complex Hadamard matrices are of Butson type with 4-th roots of unity.

Characterization of eigenmatrices for association schemes containing such matrices.

Classification of association schemes on Galois rings of characteristic 4.

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.