11 problems

Let $m$ be the parameter used by the paper's numerical method, and let the method produce an upper bound for sphere packing density. Let the optimal bound obtainable from the paper…

Let $n$ be the dimension of Euclidean space, and let the paper's direct linear-programming sphere-packing bound be applied in dimension $n$. Sharp-bound conjecture. Based on numeri…

The $E_8$ root lattice is the even unimodular lattice in $\boldsymbol{R}^8$ with covolume $1$, and the Leech lattice is the even unimodular lattice in $\boldsymbol{R}^{24}$ with no…

Let $n$ be the dimension of Euclidean space, and let a sphere packing be a collection of congruent balls with disjoint interiors. The paper constructs direct linear-programming upp…

The sphere packing problem asks for the densest packing of congruent spheres in Euclidean space. The paper develops linear-programming bounds directly for sphere packing, with part…

Distance bound conjecture. The distance of an $[[n,1,d]]$ $M_3$-code satisfies $d<d_{\max}$, with

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.

BDHSS test-function conjecture. If

In two dimensions, consider the linear programming bounds for energy minimization and the hexagonal lattice. Cohn–Kumar sharpness conjecture. The linear programming bounds should b…

Fix a dimension $d\geq 3$, density $\rho=1$, and $\alpha>0$. For the potential $p(r)=e^{-\alpha r^2}$, consider the optimal linear programming bound $\widehat{f}(0)-f(0)$ from the…

Optimality conjecture. The bound given by Theorem is the optimal solution to this linear program.