In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Rob T. Curtis
for manipulating the
Mathieu group
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objec ...
s,
binary Golay code
In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection ...
and
Leech lattice.
Description
The Miracle Octad Generator is a 4x6 array of
combinations describing any point in 24-dimensional space. It preserves all of the symmetries and
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'' s ...
s of the
Mathieu group
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objec ...
M
24, 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.
Golay code
Another use for the Miracle Octad Generator is to quickly verify codewords of the
binary Golay code
In mathematics and electronics engineering, a binary Golay code is a type of linear error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection ...
. Each element of the Miracle Octad Generator can store either a '1' or a '0', usually displayed as an
asterisk
The asterisk ( ), from Late Latin , from Ancient Greek , ''asteriskos'', "little star", is a typographical symbol. It is so called because it resembles a conventional image of a heraldic star.
Computer scientists and mathematicians often voc ...
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
Parity may refer to:
* Parity (computing)
** Parity bit in computing, sets the parity of data for the purpose of error detection
** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the r ...
. The other restriction is that the ''scores'' of each column form a word in the
hexacode In coding theory, the hexacode is a length 6 linear code of dimension 3 over the Galois field GF(4)=\ of 4 elements defined by
:H=\.
It is a 3-dimensional subspace of the vector space of dimension 6 over
GF(4).
Then H contains 45 codewords of weig ...
. 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 M
12 and
ternary Golay code
In coding theory, the ternary Golay codes are two closely related error-correcting codes.
The code generally known simply as the ternary Golay code is an 1, 6, 53-code, that is, it is a linear code over a ternary alphabet; the relative distance ...
as the Miracle Octad Generator does for M
24 and binary Golay code, respectively. Instead of using a quaternary hexacode, the MiniMOG uses a ternary tetracode.
Notes
References
*
*{{Citation , last1=Curtis , first1=R. T. , title=A new combinatorial approach to M
24 , mr=0399247 , year=1976 , journal=Mathematical Proceedings of the Cambridge Philosophical Society , issn=0305-0041 , volume=79 , issue=1 , pages=25–42 , doi=10.1017/S0305004100052075
External links
The Miracle Octad Generator
Sporadic groups