Related papers: New upper bounds for spherical codes and packings
We show that the spectral embeddings of all known triangle-free strongly regular graphs are optimal spherical codes (the new cases are $56$ points in $20$ dimensions, $50$ points in $21$ dimensions, and $77$ points in $21$ dimensions), as…
This is the sixth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…
In this paper we study the hard sphere packing problem in the Hamming space by the cavity method. We show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically…
We study the problem of compression for the purpose of similarity identification, where similarity is measured by the mean square Euclidean distance between vectors. While the asymptotical fundamental limits of the problem - the minimal…
In this short note we revisit the upper bound of the asymptotic least density of covering codes of radius $R$ in $[q]^n$ established by Krivelevich, Sudakov, and Vu. We show that by using a slightly different optimization in their core…
Pseudoline arrangements are fundamental objects in discrete and computational geometry, and different works have tackled the problem of improving the known bounds on the number of simple arrangements of $n$ pseudolines over the past…
In this paper, we study the problem of hyperball (hypersphere) packings in $n$-dimensional hyperbolic space ($n \ge 4$). We prove that to each $n$-dimensional congruent saturated hyperball packing, there is an algorithm to obtain a…
We prove upper bounds on the average kissing number $k(\mathcal{P})$ and contact number $C(\mathcal{P})$ of an arbitrary finite non-congruent sphere packing $\mathcal{P}$, and prove an upper bound on the packing density…
Let the kissing number $K(d)$ be the maximum number of non-overlapping unit balls in $\mathbb R^d$ that can touch a given unit ball. Determining or estimating the number $K(d)$ has a long history, with the value of $K(3)$ being the subject…
In the early 1990s, A. Bezdek and W. Kuperberg used a relatively simple argument to show a surprising result: The maximum packing density of circular cylinders of infinite length in $\mathbb{R}^3$ is exactly $\pi/\sqrt{12}$, the planar…
We establish an improved upper bound for the number of incidences between m points and n circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension $\ge 2$, is…
Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for…
Define the superball with radius $r$ and center ${\boldsymbol 0}$ in $\mathbb{R}^n$ to be the set $$ \left\{{\boldsymbol x}\in\mathbb{R}^n:\sum_{j=1}^{m}\left(x_{k_j+1}^2+x_{k_j+2}^2+\cdots+x_{k_{j+1}}^2\right)^{p/2}\leq…
The densest local packing (DLP) problem in d-dimensional Euclidean space Rd involves the placement of N nonoverlapping spheres of unit diameter near an additional fixed unit-diameter sphere such that the greatest distance from the center of…
The densest local packings of N identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained using a nonlinear programming method operating in conjunction with a stochastic search of…
We present a new lower bound on the Bowen-Radin maximal density of radius-R ball packings in the m-dimensional hyperbolic space, improving on the basic covering bound by factor \Omega(m(R+\ln m)) as m tends to infinity. This is done by…
The packing of hard spheres (HS) of diameter $\sigma$ in a cylinder has been used to model experimental systems, such as fullerenes in nanotubes and colloidal wire assembly. Finding the densest packings of HS under this type of confinement,…
The problem of packing a system of particles as densely as possible is foundational in the field of discrete geometry and is a powerful model in the material and biological sciences. As packing problems retreat from the reach of solution by…
Inspired by the boolean discrepancy problem, we study the following optimization problem which we term \textsc{Spherical Discrepancy}: given $m$ unit vectors $v_1, \dots, v_m$, find another unit vector $x$ that minimizes $\max_i \langle x,…
In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We apply this to get sharp bounds for packing problems,…