# Twenty-three constructions for the Leech lattice

@article{Conway1982TwentythreeCF, title={Twenty-three constructions for the Leech lattice}, author={John H. Conway and N. J. A. Sloane}, journal={Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences}, year={1982}, volume={381}, pages={275 - 283} }

In a recent paper with R. A. Parker we classified the points at maximum distance from the Leech lattice (the ‘deepest holes’ in that lattice), and showed that there are 23 classes of such holes, the classes being in one: one correspondence with the 23 Niemeier lattices in 24 dimensions. We now present 23 constructions for the Leech lattice, one for each class of hole or Niemeier lattice. Two of these are the usual constructions of the Leech lattice from the Golay codes over GF(2) and GF(3).

## 12 Citations

The Leech lattice and other lattices

- Mathematics
- 1999

This is an unpublished manuscript written in 1983-4. It contains several results about lattices (=integral quadratic forms) including the classification of the unimodular lattices in dimensions up to…

Low-dimensional lattices V. Integral coordinates for integral lattices

- MathematicsProceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1989

We say that an n-dimensional (classically) integral lattice ⋀ is s-integrable, for an integer s, if it can be described by vectors s-½(x1,...,xk), with all xi ∊ Z, in a euclidean space of dimension k…

Sphere Packings, Lattices and Groups

- MathematicsGrundlehren der mathematischen Wissenschaften
- 1988

The second edition of this book continues to pursue the question: what is the most efficient way to pack a large number of equal spheres in n-dimensional Euclidean space? The authors also continue to…

More efficient bounded-distance decoding of the Golay code and the Leech lattice

- Computer ScienceProceedings of 1994 IEEE International Symposium on Information Theory
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It is shown that the 'holy construction' of the Leech lattice with theOctacode as the glue code is essentially different from the permuted Turyn construction, although both constructions rely on the octacode.

Beauty and the beast: Superconformal symmetry in a monster module

- Mathematics
- 1988

Frenkel, Lepowsky, and Meurman have constructed a representation of the largest sporadic simple finite group, the Fischer-Griess monster, as the automorphism group of the operator product algebra of…

Geometry of exceptional super Yang-Mills theories

- MathematicsPhysical Review D
- 2019

Some time ago, Sezgin, Bars and Nishino have proposed super Yang-Mills theories (SYM's) in $D=11+3$ and beyond. Using the "Magic Star" projection of $\mathfrak{e}_{8(-24)}$, we show that the…

Quantum Symmetries: From Clifford and Hurwitz Algebras to M-Theory and Leech Lattices

- MathematicsAdvances in Applied Clifford Algebras
- 2018

We explore some consequences of a theory of internal symmetries for elementary particles based on exceptional quantum mechanical spaces based on Jordan algebra formulation that admit exceptional…

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