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 $4n^2$ 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 $4$.

The Multiplicities of a Dual-thin $Q$-polynomial Association Scheme

Let $Y=(X, \{ R_i \}_{1\le i\le D})$ denote a symmetric association scheme, and assume that $Y$ is $Q$-polynomial with respect to an ordering $E_0,...,E_D$ of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities $m_i$ $(0 \leq i \leq D)$ of $Y$ is unimodal. Talking to Terwilliger, Stanton made the related conjecture that $m_i \leq m_{i+1}$ and $m_i \leq m_{D-i}$ for $i < D/2$. We prove that if $Y$ is dual-thin in the sense of Terwilliger, then the Stanton conjecture is true.