In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by for manipulating the Mathieu group

Mathieu group

In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...

s, binary Golay code

Binary Golay code

In mathematics and electronics engineering, a binary Golay code is a type of error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics....

 and Leech lattice

Leech lattice

In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...

.

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

Mathieu group

In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...

 M₂₄, despite its simplicity. More specifically, it preserves the maximal subgroup

Maximal subgroup

In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words H is a maximal element of the partially...

s 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

Binary Golay code

In mathematics and electronics engineering, a binary Golay code is a type of error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics....

. Each element of the Miracle Octad Generator can store either a '1' or a '0', usually displayed as an asterisk

Asterisk

An asterisk is a typographical symbol or glyph. It is so called because it resembles a conventional image of a star. Computer scientists and mathematicians often pronounce it as star...

 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

John Horton Conway

John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...

 developed a 4 × 3 array known as the MiniMOG. The MiniMOG provides the same function for the Mathieu group M₁₂ and ternary Golay code

Ternary Golay code

There are two closely related error-correcting codes known as ternary Golay codes. The code generally known simply as the ternary Golay code is a perfect [11, 6, 5] ternary linear code; the extended ternary Golay code is a [12, 6, 6] linear code obtained by adding a zero-sum check digit to the...

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.