# Optimality and uniqueness of the Leech lattice among lattices

@article{Cohn2004OptimalityAU, title={Optimality and uniqueness of the Leech lattice among lattices}, author={Henry Cohn and Abhinav Kumar}, journal={arXiv: Metric Geometry}, year={2004} }

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 exceed the Leech lattice's density by a factor of more than 1+1.65*10^(-30), and we give a new proof that E_8 is the unique densest lattice in R^8.

## 123 Citations

On a generalization of Craig lattices

- Mathematics
- 2013

In this paper we introduce generalized Craig lattices, which allows us to construct lattices in Euclidean spaces of many dimensions in the range 3332 − 4096 which are denser than the densest known…

Perfect, strongly eutactic lattices are periodic extreme

- Mathematics
- 2010

Abstract We introduce a parameter space for periodic point sets, given as unions of m translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive…

Perfect, strongly eutactic lattices are periodic extreme

- Mathematics
- 2008

We introduce a parameter space for periodic point sets, given as unions of m translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive…

Towards a proof of the 24-cell conjecture

- Mathematics
- 2017

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…

Lattice packings through division algebras

- Mathematics
- 2021

In this text, we will show the existence of lattice packings in a family of dimensions by employing division algebras. This construction is a generalization of Venkatesh’s lattice packing result [1].…

On the density of cyclotomic lattices constructed from codes

- Mathematics
- 2017

Recently, Venkatesh improved the best known lower bound for lattice sphere packings by a factor loglog n for infinitely many dimensions n. Here, we prove an effective version of this result, in the…

On lattices whose minimal vectors form a 6-design

- Computer Science, MathematicsEur. J. Comb.
- 2009

Let L be a lattice of dimension [email protected]?24 such that the minimal vectors of L form a 6-design and generate L. Then L is similar to either the root lattice E"8, the Barnes-Wall lattice…

A Note on Lattice Packings via Lattice Refinements

- Mathematics, Computer ScienceExp. Math.
- 2018

A simple o(nn/2) running time algorithm that refines successively the packing lattice Dn (checkboard lattice) of the unit ball Bn and terminates with packing lattices achieving the best-known lattice densities.

Some properties of optimal functions for sphere packing in dimensions 8 and 24

- Mathematics
- 2016

We study some sequences of functions of one real variable and conjecture that they converge uniformly to functions with certain positivity and growth properties. Our conjectures imply a conjecture of…

A Mordell inequality for lattices over maximal orders

- Mathematics, Computer Science
- 2008

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…

## References

SHOWING 1-10 OF 44 REFERENCES

Uniqueness of Certain Spherical Codes

- Mathematics
- 1981

We show that there is essentially only one way of arranging 240 (resp. 196560) nonoverlapping unit spheres in R 8 (resp.R 24) so that they all touch another unit sphere Ω n , and only one way of…

New upper bounds on sphere packings I

- Mathematics
- 2003

We continue the study of the linear programming bounds for sphere packing introduced by Cohn and Elkies. We use theta series to give another proof of the principal theorem, and present some related…

New upper bounds on sphere packings II

- Mathematics
- 2002

We continue the study of the linear programming bounds for sphere packing introduced by Cohn and Elkies. We use theta series to give another proof of the principal theorem, and present some related…

From error-correcting codes through sphere packings to simple groups

- Mathematics
- 1983

1. The origin of error-correcting codes an introduction to coding the work of Hamming the Hamming-Holbrook patent the Hamming codes are linear the work of Golay the priority controversy 2. From…

SPHERICAL CODES AND DESIGNS

- Mathematics
- 1991

Publisher Summary This chapter provides an overview of spherical codes and designs. A finite non-empty set X of unit vectors in Euclidean space R d has several characteristics, such as the dimension…

The complete enumeration of extreme senary forms

- MathematicsPhilosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
- 1957

Let f(x1..., x6)be a positive definite senary quadratic form of determinant D. Let M be its minimum value for integers x1..., x6, not all zero. The form is said to be extreme if, for all…

Bounds for unrestricted codes, by linear programming

- Mathematics
- 1972

The paper describes a problem of linear programming associated with distance properties of unrestricted codes. As a solution to the problem, one obtains an .upper bound for the number of words in…

Matrix analysis

- Computer Science, Mathematics
- 1985

This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.

Methods and Structure of Commutative Harmonic Analysis

- Mathematics
- 1991

Many really significant “final” scientific achievements share the following two characteristics. First of all, they are sufficiently trivial, and, hence, can become of “common use”, i.e., necessary…

New Bounds on the Number of Unit Spheres That Can Touch a Unit Sphere in n Dimensions

- Computer Science, MathematicsJ. Comb. Theory, Ser. A
- 1979

Abstract New upper bounds are given for the maximum number, τ n , of nonoverlapping unit spheres that can touch a unit sphere in n -dimensional Euclidean space, for n ⩽24. In particular it is shown…