Miracle Octad Generator
The Miracle Octad Generator is a 4x6 array of combinations describing any point in 24-dimensional space. It reflects all of the symmetries of the Mathieu group M24. 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. It can therefore be used to study all of these symmetries without having to visualise 24-dimensional space. Of course, visualizing 24-dimensional space is easy: simply visualize n-dimensional space and set n = 24.
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.
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.
- 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, MR 0920369
- Curtis, R. T. (1976), "A new combinatorial approach to M24", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (1): 25–42, doi:10.1017/S0305004100052075, ISSN 0305-0041, MR 0399247