Experimental Mathematics

Orbits in the Leech Lattice

Daniel Allcock

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We provide an algorithm for determining whether two vectors in the Leech lattice are equivalent under its isometry group, the Conway group $\co0$ of order $\sim8\times10^{18}$. Our algorithm reduces the test of equivalence to at most four tests under the subgroup $2^{12}{:}M_{24}$ 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 ${}\sim24$ with good properties such as having small determinant. Our methods rely on and develop the work of R. T. Curtis.

Article information

Experiment. Math., Volume 14, Issue 4 (2005), 491-509.

First available in Project Euclid: 10 January 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20D08: Simple groups: sporadic groups 11H06: Lattices and convex bodies [See also 11P21, 52C05, 52C07]

Leech lattices S-lattices Golay code orbit


Allcock, Daniel. Orbits in the Leech Lattice. Experiment. Math. 14 (2005), no. 4, 491--509. https://projecteuclid.org/euclid.em/1136926978

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