Related papers: Dense lattices in low dimensions
We classify the dual strongly perfect lattices in dimension 16. There are four pairs of such lattices, the famous Barnes-Wall lattice $\Lambda _{16}$, the extremal 5-modular lattice $N_{16}$, the odd Barnes-Wall lattice $O_{16}$ and its…
It is well-known that the densest lattice sphere packings also typically have large kissing numbers. The sphere packing density maximization problem is known to have a solution among well-rounded lattices, of which the integer lattice…
In this research announcement we outline the methods used in our recent proof that the Leech lattice is the unique densest lattice in R^24. Complete details will appear elsewhere, but here we illustrate our techniques by applying them to…
This review paper is devoted to the problems of sphere packings in 4 dimensions. The main goal is to find reasonable approaches for solutions to problems related to densest sphere packings in 4-dimensional Euclidean space. We consider two…
We prove that the Leech lattice is the unique densest lattice in R^24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R^24 can…
1) We present new lattice sphere packings in Euclid spaces of many dimensions in the range 3332-4096, which are denser than known densest Mrodell-Weil lattice sphere packings in these dimensions. Moreover it is proved that if there were…
A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be…
In the first paper of this series, we constructed a family of lattices in dimensions 2^{n+1} for positive integers n, and proved that the associated lattice packings of spheres equal or exceed the previous records for several values of n.…
In 1908, Voronoi introduced an algorithm that solves the lattice packing problem in any dimension in finite time. Voronoi showed that any lattice with optimal packing density must be a so-called perfect lattice, and his algorithm enumerates…
In this paper we prove an analogue of Mordell's inequality for lattices in finite-dimensional complex or quaternionic Hermitian space that are modules over a maximal order in an imaginary quadratic number field or a totally definite…
Particle packing problems have fascinated people since the dawn of civilization, and continue to intrigue mathematicians and scientists. Resurgent interest has been spurred by the recent proof of Kepler's conjecture: the face-centered cubic…
We study statistical and structural properties of extreme lattices, which are the local minima in the density landscape of lattice sphere packings, in $d$-dimensional Euclidean space $\mathbb{R}^d$. Specifically, we ascertain the…
We give classifications of integral lattices which include the Barnes-Wall lattice $BW_{16}$ or laminated lattices of dimension $1 \leqslant d \leqslant 8$ and of minimum 4. Also, we give certain lattice neighboring from each lattice.…
We prove that the kissing numbers in 17, 18, 19, 20, and 21 dimensions are at least 5730, 7654, 11692, 19448, and 29768, respectively. The previous records were set by Leech in 1967, and we improve on them by 384, 256, 1024, 2048, and 2048.…
Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic…
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…
We use the automorphism group $Aut(H)$, of holes in the lattice $L_8=A_2\oplus A_2\oplus D_4$, as the starting point in the construction of sphere packings in 10 and 12 dimensions. A second lattice, $L_4=A_2\oplus A_2$, enters the…
In this note we give a simple proof of the classical fact that the hexagonal lattice gives the highest density circle packing among all lattices in $R^2$. With the benefit of hindsight, we show that the problem can be restricted to the…
How many unit $n-$dimensional spheres can simultaneously touch or kiss a central $n-$dimensional unit sphere? Beyond mathematics this question has implications for fields such as cryptography and the structure of biologic and chemical…
We present an efficient Monte Carlo method for the lattice sphere packing problem in d dimensions. We use this method to numerically discover de novo the densest lattice sphere packing in dimensions 9 through 20. Our method goes beyond…