25 problems

Let $C$ be a PULB-optimal code, and let $D$ be its set of universal minima. Assume that $D$ is in general position, meaning that $D$ is not contained in a hyperplane. Universal pol…

Let $-1 \leq \beta < 0 \leq \alpha < 1$. A spherical two-distance set in $\mathbb{R}^d$ is a collection of unit vectors whose pairwise inner products lie in $\{\alpha,\beta\}$. Let…

Cohn–de Laat–Leijenhorst conjecture. Three-point semidefinite programming bounds prove that $C$ is a maximal spherical code.

Let $\mathcal{C}$ be the set of $24$ vertices of a snub cube, regarded as unit vectors in $\mathbb{R}^3$. Snub-cube optimality conjecture. Three-point bounds prove that…

For each integer $k e 0$, let a Kerdock spherical code be the spherical code in $\mathbb{R}^{2^{2k}}$ constructed from the Kerdock binary code, with $2^{4k}+2^{2k+1}$ points and…

Let $\alpha,\beta \in (0,1)$ and let $L \subseteq [-1,-\beta]\cup\{\alpha\}$ with $\alpha \in L$. An $L$-code is a finite set of unit vectors in Euclidean space whose pairwise inne…

Let $\mathbb{S}^{d-1}$ be the unit sphere, and consider codebooks consisting of points on this sphere. A simplex codebook is a maximally separated codebook; the paper also consider…

Folklore conjecture. Let $C\subset\mathbb{S}^{n-1}$, with $n\geq 2$, be a spherical 4-distance 7-design. Then $C$ is a tight spherical 7-design. In particular,

Bannai et al.'s classification conjecture. One of the following holds: (1) $n=2$ and $C$ is a regular hexagon or a regular heptagon; (2) $C$ is a tight spherical 5-design; or (3)…

Fix $-1\le\beta<0\le\alpha<1$. For a spherical two-distance set in $\mathbb R^d$ with inner products in $\{\alpha,\beta\}$, let $N_{\alpha,\beta}(d)$ be the maximum possible size.…

Strengthened Gerzon bound. Then

Let $\mathcal C$ be the code constructed in the paper consisting of $85$ points in $\mathbb C^5$, viewed projectively. A code is universally optimal if it minimizes every admissibl…

Interval conjecture. For every dimension $n$, the values $N\in [D(n,\tau),D(n,\tau+1)]$ for which second-level bounds exist form an interval.

The $q$-moment conjecture. If $\mu$ is an isotropic probability mass on $\mathbb{R}^k$, then

For $N\geq 4$, let $r_{\max}(N)$ be the maximal radius of $N$ equal spheres touching a central unit sphere. Robinson's conjecture. For all $N\geq 4$, one has … except possibly for…

Let $v_1,\dots,v_6$ be unit vectors in $\mathbf{R}^3$, and suppose one of the vectors lies at the north pole. An optimal configuration maximizes the minimum absolute determinant amo…

Let $\beta\in(0,1)$ be fixed, let $\alpha_1,\dots,\alpha_k$ be any $k$ real numbers, and let a spherical $L$-code be a finite non-empty set of unit vectors in $\mathbb{R}^n$ whose…

Let $S^2$ be the unit sphere, and let a completely monotonic potential function be a potential function whose derivatives alternate in sign on the relevant domain. For five points…

Polynomial spherical-code conjecture. Suppose $\alpha_1,\dotsc,\alpha_k$ are any $k$ real numbers. Then

Let $N_{1/(2r-1)}(d)$ denote the maximum number of equiangular lines in $\mathbb{R}^d$ with common angle $\arccos \frac{1}{2r-1}$, where $r\geq 2$ is an integer. Asymptotic equiang…

Local dimension conjecture. In a neighborhood of the explicitly constructed code, the space of tight $39$-point codes modulo the action of $F_4$ is a manifold of dimension $24$.

Local dimension conjecture. There exists a $24$-point, respectively $25$-point, tight simplex in ${\mathbb O}{\mathbb P}^2$ such that, in a neighborhood of it, the space of tight s…

Local dimension conjecture. There exists a $12$-point, respectively $13$-point, tight simplex in ${\mathbb H}{\mathbb P}^2$ such that, in a neighborhood of it, the space of tight s…

Existence conjecture. As $d\to\infty$, there exist tight $N$-point simplices in ${\mathbb H}{\mathbb P}^{d-1}$ for every $N$ satisfying

Square-antiprism conjecture. For every completely monotonic potential function $f$ on $(0,4]$, some square antiprism minimizes the energy $E_f$ among all eight-point codes in $S^2$…