11 problems

Classification conjecture. Any MUB-triplet $(X_1,X_2,X_3)$ in dimension $d=6$ is permutationally unitary equivalent to an MUB-triplet obtained by the construction $(\mathrm{hff})$.…

Let $D$ be an arbitrary dephased, symmetric complex Hadamard matrix arising from the conference matrix construction, of order $N$. Apply Construction by repeatedly selecting suitab…

Szöllősi-family non-extendability conjecture. Any matrix $S\in X(\alpha)$ in the Szöllősi family cannot be part of an MUB-quadruplet.

Non-isolatedness conjecture. Only non-isolated complex Hadamard matrices of order $N=d^2$ provide two-unitary structures.

Small-size classification conjecture. The matrices in Proposition 1.8 are the only $4\times4$ Hadamard matrices, and there is no $5\times5$ Hadamard matrix with self-adjoint entrie…

Let $C_n^{\operatorname{circ}}(\infty)$ denote the set of circulant complex Hadamard matrices of order $n$, with no fixed bound on the orders of their root-of-unity entries, and le…

Main conjecture. For $H\in C_N(s)$, with $s\in\mathbb N$ chosen to be minimal, $d(H)$ can be recaptured from the knowledge of the associated measure $\mu$.

Measure-theoretic defect conjecture. For $H\in C_N(s)$, with $s$ minimal,

Defect bound conjecture. We have

Rational defect conjecture. For the Butson matrices we have $d_{\mathbb Q}(H)=d(H)$.

Maximal-family conjecture. The maximal affine family of complex Hadamard matrices stemming from the Fourier matrix $F_d$ in dimensions $d=2p$, where $p$ is prime, is given by Eqs.…