41 problems

Let $M(n)$ denote the maximum number of pairwise mutually unbiased bases in $\mathbb{C}^n$. For a prime power $n=p^r$, one has $M(p^r)=p^r+1$; the unresolved case concerns dimensio…

Defect bound conjecture. We have

A complex Hadamard matrix of dimension $N$ is a matrix whose entries have modulus one and whose rows are pairwise orthogonal. Two such matrices are equivalent if one can be obtaine…

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…

Let $H_8$ be the order-$8$ real Hadamard matrix, and let the parametrized complex Hadamard matrices constructed by Theorem b be the families obtained from the construction describe…

Let $H(\theta)$ be the one-parameter family of self-adjoint complex Hadamard matrices of order $6$ defined above, with…

Let $T^{(3)}_{8C}$ denote the third family of complex Hadamard matrices of order $8$, obtained from the family associated with the template $T_{8C}$ and its solution branches. The…

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

Hadamard-cube vanishing conjecture. For every $\pi\in S_6$ and every $j=1,2,3$,

Eigenvalue–eigenvector classification conjecture. The eigenvalues and eigenvectors of $6\times 6$ CHMs should lead to their complete classification.

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

Nonexistence conjecture for four MUBs in dimension six. Four mutually unbiased bases should not exist in $\bbC^6$.

Let $d$ be the dimension of a SIC, and consider the projector, complex Hadamard matrix, equiangular tight frame, and, where applicable, tight fusion frame structures constructed fr…

Let an $\operatorname{HFP}(2t,4_{\mathbf u})$-code be a Hadamard full propelinear code of the indicated type and length $4t$. HFP-code nonexistence conjecture. There do not exist…

A circulant complex Hadamard matrix is a complex Hadamard matrix whose rows are cyclic shifts of one another. Arasu–de Launey–Ma conjecture. There is no circulant complex Hadamard…

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 $D_6$ be the dephased order-six Hadamard matrix considered in the source, and let the center manifold be parametrized by coordinates $t_1,\ldots,t_4$ associated with the chosen…

Let $F_6$ be the order-six Fourier matrix. Let $\alpha_i$ be the time-rate functions for the four center-manifold coordinates in the embedding used by the source. Four-dimensional…

Let $F_6$ be the order-six Fourier matrix, and let the defect of a complex Hadamard matrix denote the dimension of the corresponding linearized solution space used in the source. A…

Let $S_6^{(0)}$ be the isolated $6\times 6$ Hadamard matrix, and let Szöllősi's generic four-parameter family be the family of order-six Hadamard matrices constructed by his genera…

Englert's conjecture. The states above trace out the curve of minimal entropies. The conjecture is correct for $d=2$, appears to hold numerically for $d=3$, and fails for $d=4$, wh…

Let $H$ be a complex Hadamard matrix of order $n$, and let a trade be a set of entries of $H$ that can be changed to obtain a different complex Hadamard matrix. Minimum-size trade…

Dimension-six linear-programming constraint. Every complex Hadamard matrix in dimension $6$, except the Spectral Hadamard, must satisfy this additional constraint.

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 $\mu$.