Orbits in the Leech Lattice

Abstract

We provide an algorithm for determining whether two vectors in the Leech lattice are equivalent under its isometry group, the Conway group Co ₀ of order ~ 8×10¹⁸. Our algorithm reduces the test of equivalence to at most four tests under the subgroup 2¹²:M ₂₄ and a test under this subgroup to at most 12 tests under M ₂₄. We also give algorithms for testing equivalence under these two subgroups. We describe our intended applications to the symmetry groups of Lorentzian lattices and the enumeration of lattices of dimension ~ 24 with good properties such as having small determinant. Our methods rely on and develop the work of R. T. Curtis.

Keywords:

• Primary 20D08, 11H06
• Leech lattices
• S-lattices
• Golay code
• orbit