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A353294
A generator matrix for the Leech lattice, multiplied by sqrt(8), read by rows.
1
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
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.
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
KEYWORD
sign,tabf,fini,full
AUTHOR
Paolo Xausa, Apr 12 2022
STATUS
approved