A353294 A generator matrix for the Leech lattice, multiplied by sqrt(8), read by rows.

8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,1

Comments

There are infinitely many such matrices, this just happens to be a concrete example that we gave in the Sphere-Packing book. It is not unique in any way. - N. J. A. Sloane, Jun 04 2022

Links

Paolo Xausa, Table of n, a(n) for n = 1..576 (rows 1..24 of the matrix, flattened)

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd edition, Springer, New York, NY, 1999, pp. 131-133.

Wikipedia, Leech lattice.

Wikipedia, Miracle Octad Generator.

Formula

det(A/sqrt(8)) = 1, where A is the present matrix.

Example

As depicted by Conway and Sloane (1999), p. 133, the full 24 X 24 matrix is given below, in standard MOG (Miracle Octad Generator) coordinates.
.
   8 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 4 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 4 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 0 4 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   --------|---------|---------|---------|---------|--------
   4 0 0 0 | 4 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 0 0 | 0 4 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 0 0 | 0 0 4 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   2 2 2 2 | 2 2 2 2 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   --------|---------|---------|---------|---------|--------
   4 0 0 0 | 0 0 0 0 | 4 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 0 0 | 0 0 0 0 | 0 4 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   4 0 0 0 | 0 0 0 0 | 0 0 4 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   2 2 2 2 | 0 0 0 0 | 2 2 2 2 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0
   --------|---------|---------|---------|---------|--------
   4 0 0 0 | 0 0 0 0 | 0 0 0 0 | 4 0 0 0 | 0 0 0 0 | 0 0 0 0
   2 2 0 0 | 2 2 0 0 | 2 2 0 0 | 2 2 0 0 | 0 0 0 0 | 0 0 0 0
   2 0 2 0 | 2 0 2 0 | 2 0 2 0 | 2 0 2 0 | 0 0 0 0 | 0 0 0 0
   2 0 0 2 | 2 0 0 2 | 2 0 0 2 | 2 0 0 2 | 0 0 0 0 | 0 0 0 0
   --------|---------|---------|---------|---------|--------
   4 0 0 0 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 | 4 0 0 0 | 0 0 0 0
   2 0 2 0 | 2 0 0 2 | 2 2 0 0 | 0 0 0 0 | 2 2 0 0 | 0 0 0 0
   2 0 0 2 | 2 2 0 0 | 2 0 2 0 | 0 0 0 0 | 2 0 2 0 | 0 0 0 0
   2 2 0 0 | 2 0 2 0 | 2 0 0 2 | 0 0 0 0 | 2 0 0 2 | 0 0 0 0
   --------|---------|---------|---------|---------|--------
   0 2 2 2 | 2 0 0 0 | 2 0 0 0 | 2 0 0 0 | 2 0 0 0 | 2 0 0 0
   0 0 0 0 | 0 0 0 0 | 2 2 0 0 | 2 2 0 0 | 2 2 0 0 | 2 2 0 0
   0 0 0 0 | 0 0 0 0 | 2 0 2 0 | 2 0 2 0 | 2 0 2 0 | 2 0 2 0
  -3 1 1 1 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1

Crossrefs

Cf. A008408, A260646, A351831.

Sequence in context: A128820 A067485 A180225 * A127886 A270033 A085121

Adjacent sequences: A353291 A353292 A353293 * A353295 A353296 A353297

Keyword

sign,tabf,fini,full

Author

Paolo Xausa, Apr 12 2022

Status

approved