72 problems

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…

Let $N_{[-1,\beta]\cup\{\alpha_1,\dots,\alpha_k\}}(d)$ denote the maximum size of a spherical two-distance set in $\mathbb R^d$ whose inner products lie in…

Let $f(r)=-\log r$ be the logarithmic potential, and consider configurations of $11$ points on $S^3$. A flat torus in $S^3$ is the product of two circles of radius $1/\sqrt{2}$ in…

View $\mathbb{R}^4$ as the orthogonal direct sum of two planes, and let $T\subset S^3$ be the flat two-dimensional torus given by the product of the circles of radius $1/\sqrt{2}$…

Let $C$ be the described $40$-point code on the unit sphere in $\mathbb{R}^{10}$, whose maximal inner product is $1/6$. Uniqueness conjecture for the 40-point code. This code is th…

Let $\mathcal{C}_n\subset S^{n-1}$ be the spherical code consisting of one north pole and two dual $n$-point simplices, with $\alpha$ the unique root in $(0,1/n)$ of … The cosine o…

Classification conjecture for 24-point spherical 5-designs. There are no 24-point spherical 5-designs in $S^3$ other than those in the three-dimensional family of deformations of…

Nonexistence conjecture for universally optimal 24-point codes. There is no universally optimal spherical code of 24 points in $S^3$.

For $n\geq 3$, a universal optimum in $\mathbb{R}^n$ is a point configuration minimizing energy for every completely monotonic potential function. Finiteness conjecture. For each…

A sharp configuration is a configuration for which the linear programming bound for spherical codes is attained. The techniques conjecture. The techniques developed in the paper ap…

Bezdek's conjecture.

Let $S$ be the Smith $(10,40,1/6)$ spherical code, a set of $40$ points on the $9$-sphere with minimum angle parameter $1/6$. A potential function on a sphere is completely monoton…

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…

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

Let $S^3$ be the unit sphere in $\mathbb{R}^4$. A spherical code is a finite subset of $S^3$, and it is universally optimal if it minimizes the energy … for every absolutely monoto…

Consider the two spherical codes in the source having respectively $40$ points in $\mathbb{R}^{10}$ and $64$ points in $\mathbb{R}^{14}$. Universal-optimality conjecture. These two…

Let Table denote the table of candidate spherical codes discussed in the source, consisting of codes with transitive symmetry groups. Small-code optimality conjecture. All the 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 an optimal codebook consist of $n+1$ codewords in $n$-dimensional Euclidean space, subject to a per-codeword energy constraint. A regular simplex is the $n$-dimensional general…

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…

Cubic-root growth conjecture. The order of growth of $r\,\rho(r,2r+k)$ is

Let $s_{2n}$ denote the best covering radius of a configuration of $2n$ points on $S^{n-1}$, and let $s_{2,2n}$ and $s_{3,2n}$ denote the corresponding covering-radius quantities a…

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,