Twenty-three constructions for the Leech lattice

  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},
  pages={275 - 283}
  • J. Conway, N. Sloane
  • Published 1982
  • Mathematics
  • Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
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). 

Figures and Tables from this paper

The Leech lattice and other lattices
This is an unpublished manuscript written in 1983-4.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
Octonions and the Leech lattice
Low-dimensional lattices V. Integral coordinates for integral lattices
  • J. Conway, N. Sloane
  • Mathematics
    Proceedings 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
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
  • F. Sun, H. van Tilborg
  • Computer Science
    Proceedings of 1994 IEEE International Symposium on Information Theory
  • 1994
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
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
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
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


A characterisation of Leech's lattice
[-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
A Setting for the Leech Lattice
Sphere Packings and Error-Correcting Codes
Error-correcting codes are used in several constructions for packings of equal spheres in n-dimensional Euclidean spaces En. These include a systematic derivation of many of the best sphere packings
Notes on sphere packings
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
On the classification of integral even unimodular 24 - dimensional quadratic forms
  • Proc . Steklov Inst . Math
  • 1980