# Notes on Sphere Packings

@article{Leech1967NotesOS, title={Notes on Sphere Packings}, author={John Leech}, journal={Canadian Journal of Mathematics}, year={1967}, volume={19}, pages={251 - 267} }

These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have a further digit added; thus §1.41 here supplements §1.4 of (4). References by section numbers are always to (4) or to the present notes, but references to other papers are numbered independently. The principal results of these notes are the following. New sphere packings are given in [2 m ], m ⩾ 6, and in [24], which are twice as…

## 176 Citations

FURTHER LATTICE PACKINGS IN HIGH DIMENSIONS

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Barnes and Sloane recently described a "general construction" for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and…

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The first construction, Construction D, resembles Construction C in that it is also based on a sequence of codes, but differs in producing lattices provided only that the codes are binary, linear and nested.

NEW SPHERE PACKINGS IN DIMENSIONS 9-15 BY JOHN LEECH AND N. J. A. SLOANE

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The packings obtained are nonlattice packings using nonlinear single error-correcting codes that have more codewords than any comparable group code, and are more densely packed than any group code of the same length and minimum distance.

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Since Minkowski [29] gave his famous lattice point theorem for centrally symmetric convex bodies, a theorem that turned out to be of fundamental importance in number theory, much effort has been made…

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This work addresses the problem of covering Rn with congruent balls, while minimizing the number of balls that contain an average point, and gives a closed formula for the covering density that depends on the distortion parameter.

Symmetric representation of the elements of the Conway group .0

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A characterisation of Leech's lattice

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[-5, 6] promises to be the subject of many investigations. We give here a short proof that this lattice is characterised by some of its simplest properties. Although we must quote two theorems to…

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Let K be a number field, and let R be the ring of algebraic integers in K. We say that K is Euclidean, or that R is Euclidean with respect to the norm. if for every a, beR, bq=O, there exist c, deR…

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Let X ⊂ Rm be a spherical code (i.e., a finite subset of the unit sphere) and consider the ideal of all polynomials in m variables which vanish on X. Motivated by a study of cometric (Q-polynomial)…

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