Miracle Octad Generator

In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Curtis (1976) for manipulating the Mathieu groups, binary Golay code and Leech lattice.

Description

The Miracle Octad Generator is an array of coordinates, arranged in four rows and six columns, capable of describing any point in 24-dimensional space. It is remarkable in the fact that it reflects all of the symmetries of the Mathieu group M₂₄, despite its simplicity. More specifically, it preserves the maximal subgroups of M₂₄, namely the monad group, duad group, triad group, octad group, octern group, sextet group, trio group and duum group. This makes it invaluable, as it can be used to study all of these symmetries without having to visualise 24-dimensional space.

Golay code

Another use for the Miracle Octad Generator is to quickly verify codewords of the binary Golay code. Each element of the Miracle Octad Generator can store either a '1' or a '0', usually displayed as an asterisk and blank space, respectively. Each column and the top row have a property known as the count, which is the number of asterisks in that particular line. One of the criteria for a set of 24 coordinates to be a codeword in the binary Golay code is for all seven counts to be of the same parity. The other restriction is that the scores of each column form a word in the hexacode. The score of a column can be either 0, 1, ω, or ω-bar, depending on its contents. The score of a column is evaluated by the following rules:

• If a column contains exactly one asterisk, it has a score of 0 if it resides in the top row, 1 if it is in the second row, ω for the third row, and ω-bar for the bottom row.
• Simultaneously complementing every bit in a column does not affect its score.
• Complementing the bit in the top row does not affect its score, either.

A codeword can be derived from just its top row and score, which proves that there are exactly 4096 codewords in the binary Golay code.

MiniMOG

John Horton Conway developed a 4 × 3 array known as the MiniMOG. The MiniMOG provides the same function for the Mathieu group M₁₂ and ternary Golay code as the Miracle Octad Generator does for M₂₄ and binary Golay code, respectively. Instead of using a quaternary hexacode, the MiniMOG uses a ternary tetracode.

References

• Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, 290 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98585-5, MR0920369 
• Curtis, R. T. (1976), "A new combinatorial approach to M₂₄", Mathematical Proceedings of the Cambridge Philosophical Society 79 (1): 25–42, ISSN 0305-0041, MR0399247