17 problems

Unique-norm integral Fourier matrix conjecture. Then

Square-free Fourier principal-minor conjecture. If $N$ is square-free, then all principal minors of ${\mathcal{F}}_N$ are nonzero.

Let $F_n$ be the Fourier matrix of order $n$, and let $\mathbb Z_n^*$ denote the multiplicative group modulo $n$. A matrix is principally non-singular if every principal submatrix…

Column-permutation conjecture. For every natural number $N$, there exists a permutation $\sigma$ such that the Fourier matrix with permuted columns has no zero principal minors.

Caragea–Lee conjecture. If $N$ is square-free, then every principal minor of $\mathcal F_N$ is nonzero.

Let $q\geq3$ be prime, let $K\in M_{M\times N}(\mathbb C)$ be an isolated truncation of $F_p$ with $p$ prime, and let $S,T$ be the sets used in Theorem 4.4. That theorem constructs…

A square master Hadamard matrix is a master Hadamard matrix of square size, and $F_p$ denotes the Fourier matrix of prime order $p$. A matrix is isolated when it has no nontrivial…

In the square case, a master Hadamard matrix is a complex Hadamard matrix arising from the master-function construction, and $F_N$ denotes the Fourier matrix of order $N$. A Dic tu…

Let $F_7$ be the Fourier matrix of order $7$, and let $P_7^q$ denote the displayed Petrescu family, with $q\in\mathbb T$. A regular complex Hadamard matrix is one whose row scalar…

Let $F_{S,p}$ denote the partial Fourier matrix associated to a subset $S\subset\mathbb Z_p$, where $p$ is prime. Truncation conjecture. There exists a constant $\varepsilon>0$ suc…

Let $s$ be an integral Fourier matrix of order $r$, and suppose that all of its entries have the same norm. Cuntz's conjecture. Then … This conjecture generalizes a result of Micha…

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…

Non-extendability conjecture. For every composite $n$, the Fourier matrix $F_n$ cannot be extended to a full set of MUHs.

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.…

Rational-span conjecture. One has

Four-parameter family conjecture. The Fourier matrix of order $6$ belongs to a smooth family of dephased Hadamard matrices with $4$ parameters.

Lusztig-family conjecture. There is a unique family ${\mathcal{F}}_C$ of $\operatorname{Irr}(W)$ such that, for some $w\in C_{\operatorname{min}}$ and some $E\in{\mathcal{F}}_C$, t…