Mutually Unbiased Bush-type Hadamard Matrices and Association Schemes

It was shown by LeCompte, Martin, and Owens in 2010 that the existence of mutually unbiased Hadamard matrices and the identity matrix, which coincide with mutually unbiased bases, is equivalent to that of a Q-polynomial association scheme of class four which is both Q-antipodal and Q-bipartite. We prove that the existence of a set of mutually unbiased Bush-type Hadamard matrices is equivalent to that of an association scheme of class five. As an application of this equivalence, we obtain an upper bound of the number of mutually unbiased Bush-type Hadamard matrices of order 4n2 to be 2n−1. This is in contrast to the fact that the best general upper bound for the mutually unbiased Hadamard matrices of order 4n2 is 2n2. We also discuss a relation of our scheme to some fusion schemes which are Q-antipodal and Q-bipartite Q-polynomial of class 4.