Mutually Unbiased Bush-type Hadamard Matrices and Association Schemes

TL;DR

This paper establishes a new equivalence between mutually unbiased Bush-type Hadamard matrices and a specific class of association schemes, providing bounds on their number and exploring related fusion schemes.

Contribution

It proves the equivalence between mutually unbiased Bush-type Hadamard matrices and class five association schemes, extending previous results and deriving new bounds.

Findings

Upper bound of 2n-1 for mutually unbiased Bush-type Hadamard matrices of order 4n^2

Equivalence between these matrices and class five association schemes

Relation to fusion schemes that are $Q$-antipodal and $Q$-bipartite $Q$-polynomial of class 4

Abstract

It was shown by LeCompte, Martin, and Oweans 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 the upper bound of the number of mutually unbiased Bush-type Hadamard matrices of order $4n^2$ to be $2n-1$. This is in contrast to the fact that the upper bound of mutually unbiased Hadamard matrices of order $4n^2$ is $2n^2$. We also discuss a relation of our scheme to some fusion schemes which are $Q$-antipodal and $Q$-bipartite $Q$-polynomial of class $4$.