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 $4n^2$ 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 $4n^2$ 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 $4$.
@article{10_37236_4915,
author = {Hadi Kharaghani and Sara Sasani and Sho Suda},
title = {Mutually unbiased bush-type {Hadamard} matrices and association schemes},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {3},
doi = {10.37236/4915},
zbl = {1325.05184},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4915/}
}
TY - JOUR
AU - Hadi Kharaghani
AU - Sara Sasani
AU - Sho Suda
TI - Mutually unbiased bush-type Hadamard matrices and association schemes
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/4915/
DO - 10.37236/4915
ID - 10_37236_4915
ER -
%0 Journal Article
%A Hadi Kharaghani
%A Sara Sasani
%A Sho Suda
%T Mutually unbiased bush-type Hadamard matrices and association schemes
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/4915/
%R 10.37236/4915
%F 10_37236_4915
Hadi Kharaghani; Sara Sasani; Sho Suda. Mutually unbiased bush-type Hadamard matrices and association schemes. The electronic journal of combinatorics, Tome 22 (2015) no. 3. doi: 10.37236/4915
