23 problems

Weak $d\times 2d$ conjecture. For every $d$, there exists a $d\times 2d$ ETF.

Let $d$ be the dimension of a real symplectic vector space, and let a $d\times n$ ETF denote an equiangular tight frame in that space. Symplectic Zauner's conjecture. There exists…

A 2-circulant ${\text{ETF}(2n,n)}$ is an equiangular tight frame of $2n$ unit vectors in $\mathbb{C}^n$ whose defining block structure is 2-circulant. Two such frames are inequival…

Let $d>3$. An equiangular tight frame (ETF) with parameters $(r,n)$ is a set of $n$ unit vectors in $\mathbb{S}^{r-1}$ whose coherence attains the Welch bound. XXY's ETF existence…

Paley ETF spark and bender conjecture. The spark of $\Phi_q$ is $((p+1)/2)^s$ and the bender is a $(p^s, ((p+1)/2)^s,\lambda)$-BIBD.

$4\times8$ uniqueness conjecture. There exists a unique complex $4\times8$ ETF up to switching equivalence.

Strong $d\times 2d$ conjecture. For each $d\neq4$, there exists a real smooth submanifold

Medium $d\times 2d$ conjecture. For every $d$, there exists a $2$-circulant $d\times 2d$ ETF.

All-dimensional ETF existence conjecture. Such equiangular tight frames exist for every $d$.

Let $\mathbb{F}=\mathbb{R}$, and let $n$ nonidentical equi-isoclinic subspaces of dimension $r$ lie in a space of dimension $d$. The real Gerzon bound is…

Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$, and let $n$ nonidentical equi-isoclinic subspaces of dimension $r$ lie in a space of dimension $d$. For…

Let $P$ be an odd prime and $J$ a positive integer. Let $(\boldsymbol{\psi}_{(\mathbf{x},\mathbf{y})})_{(\mathbf{x},\mathbf{y})\in\mathbb{F}_P^J\times\mathbb{F}_P^J}$ be the Gabor–…

Let $d\geq 5$. An equiangular tight frame (ETF) with parameters $(d,n)$ is a real ETF for $\mathbb{R}^d$ of size $n$. ETF extension conjecture. There exists an ETF with parameters…

Let $E^{\Delta}_{d,p}$ and $E_{\alpha,p}$ be the energies associated with the potentials $f^{\Delta}_{d,p}(t)=|t+\frac1d|^p$ and $f_{\alpha,p}(t)=|t^2-\alpha^2|^p$, respectively. A…

Binder BIBD conjecture. The binder of $\mathcal G(p,\ldots,p)$ is a

Let $D$ and $N$ be positive integers with $N>2D>2$, and let a real $\operatorname{ETF}(D,N)$ denote a real equiangular tight frame of $N$ vectors in dimension $D$. Call $(D,N)$ pos…

Let $D$ and $N$ be positive integers with $N>2D>2$, and let an $\operatorname{ETF}(D,N)$ denote a complex equiangular tight frame of $N$ vectors in dimension $D$. Call $(D,N)$ posi…

Mixon's divisibility conjecture. There exists an $n$-vector equiangular tight frame in $\mathbb{C}^d$ only if one of these quantities divides the product of the other two.

Minimal-vector conjecture. For every unit equiangular tight frame ${\mathcal F}$,

A Butson-type Hadamard matrix $H(p,h)$ is an $h\times h$ Hadamard matrix whose entries are all $p$th roots of unity. It is conjectured that there exists an $H(p,\lambda p)$ matrix…

Let $n$ complex unit vectors in $\mathbb{C}^m$ form an $(n,m)$-frame if they constitute a finite complex equiangular tight frame, with mutual inner products satisfying … It is know…

An equiangular tight frame (ETF) is a collection of vectors with equal pairwise inner-product magnitudes that forms a tight frame. For a matrix, the $(K,\delta)$-restricted isometr…

Let $Q$ be a complex Seidel matrix whose off-diagonal entries are $m$th roots of unity for some $m\geq 3$. Such matrices are considered up to switching equivalence, where switching…