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| | name = Extended binary Golay code
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| | image = [[File:BinaryGolayCode.svg|250px]]
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| | image_caption = Its generator matrix
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| | namesake = [[Marcel J. E. Golay]]
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| | type = [[Linear block code]]
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| | block_length = 24
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| | message_length = 12
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| | rate = 12/24 = 0.5
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| | distance = 8
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| | alphabet_size = 2
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| | notation = <math>[24,12,8]_2</math>-code
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| }}
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| {{infobox code
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| | name = Perfect binary Golay code
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| | image =
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| | image_caption =
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| | namesake = [[Marcel J. E. Golay]]
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| | type = [[Linear block code]]
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| | block_length = 23
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| | message_length = 12
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| | rate = 12/23 ~ 0.522
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| | distance = 7
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| | alphabet_size = 2
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| | notation = <math>[23,12,7]_2</math>-code
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| }}
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| In [[mathematics]] and [[electronics engineering]], a '''binary Golay code''' is a type of [[error-correcting code]] used in [[digital communication]]s. The binary Golay code, along with the [[ternary Golay code]], has a particularly deep and interesting connection to the theory of [[finite sporadic group]]s in mathematics. These codes are named in honor of [[Marcel J. E. Golay]].
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| There are two closely related binary Golay codes. The '''extended binary Golay code''' (sometimes just called the "Golay code" in finite group theory) encodes 12 bits of data in a 24-bit word in such a way that any 3-bit errors can be corrected or any 7-bit errors can be detected.
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| The other, the '''perfect binary Golay code''', has codewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position (conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a [[parity bit]]). In standard code notation the codes have parameters [24, 12, 8] and [23, 12, 7], corresponding to the length of the codewords, the [[dimension (vector space)|dimension]] of the code, and the minimum [[Hamming distance]] between two codewords, respectively.
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| == Mathematical definition ==
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| In mathematical terms, the extended binary Golay code consists of a 12-dimensional [[linear subspace|subspace]] ''W'' of the space ''V''='''F'''<sub>2</sub><sup>24</sup> of 24-bit words such that any two distinct elements of ''W'' differ in at least eight coordinates. Equivalently, any non-zero element of ''W'' has at least eight non-zero coordinates.
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| * The possible sets of non-zero coordinates as ''w'' ranges over ''W'' are called ''code words''. In the extended binary Golay code, all code words have the [[Hamming weight]]s of 0, 8, 12, 16, or 24.
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| * Up to relabeling coordinates, ''W'' is unique.
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| The perfect binary Golay code is a [[perfect code]]. That is, the spheres of radius three around code words form a partition of the vector space.
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| The [[automorphism group]] of the binary Golay code is the [[Mathieu group]] <math>M_{23}</math>.
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| The [[automorphism group]] of the extended binary Golay code is the [[Mathieu group]] <math>M_{24}</math>. The other Mathieu groups occur as [[group action|stabilizer]]s of one or several elements of ''W''.
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| The Golay [[code word]]s of weight eight are elements of the S(5,8,24) [[Steiner system]].
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| == Constructions ==
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| #[[Lexicode|Lexicographic code]]: Order the vectors in ''V'' lexicographically (i.e., interpret them as unsigned 24-bit binary integers and take the usual ordering). Starting with ''w''<sub>1</sub> = 0, define ''w''<sub>2</sub>, ''w''<sub>3</sub>, ..., ''w''<sub>12</sub> by the rule that ''w''<sub>''n''</sub> is the smallest integer which differs from all linear combinations of previous elements in at least eight coordinates. Then ''W'' can be defined as the span of ''w''<sub>1</sub>, ..., ''w''<sub>12</sub>.
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| #[[Quadratic residue code]]: Consider the set ''N'' of quadratic non-residues (mod 23). This is an 11-element subset of the [[cyclic group]] '''Z'''/23'''Z'''. Consider the translates ''t''+''N'' of this subset. Augment each translate to a 12-element set ''S''<sub>''t''</sub> by adding an element ∞. Then labeling the basis elements of ''V'' by 0, 1, 2, ..., 22, ∞, ''W'' can be defined as the span of the words ''S''<sub>''t''</sub> together with the word consisting of all basis vectors. (The perfect code is obtained by leaving out ∞.)
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| #As a [[Cyclic code]]: The perfect G<sub>23</sub> code can be constructed via factorization of <math>x^{23}-1</math>, it is the code generated by <math>x^{11}+x^{10}+x^6+x^5+x^4+x^2+1 / x^{23}-1</math>
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| #From the [[Steiner_system#The_Steiner_system_S.285.2C_8.2C_24.29| Steiner System S(5,8,24)]], consisting of 759 subsets of a 24-set. If one interprets each subset as a 0-1-codeword of length 24 (and hence Hamming-weight 8), these are the "octads" in the binary Golay code. The entire Golay code can be obtained be taking repeatetly [[symmetric difference]]s of subsets, i.e. binary addition. An easier way to write down the Steiner system resp. the octads is the [[Miracle Octad Generator]] of R. T. Curtis, that uses a particular 1:1-correspondence between the 35 partitions of an 8-set and the 35 partitions of the finite vector space <math>\mathbb{F}_2^4</math> into 4 planes. <ref>http://finitegeometry.org/sc/24/MOG.html</ref> Nowadays often the compact approach of Conway's hexacode, that uses a 4×6 array of square cells, is used.
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| #Winning positions in the [[mathematical game]] of Mogul: a position in Mogul is a row of 24 coins. Each turn consists of flipping from one to seven coins such that the leftmost of the flipped coins goes from head to tail. The losing positions are those with no legal move. If heads are interpreted as 1 and tails as 0 then moving to a codeword from the extended binary Golay code guarantees it will be possible to force a win.
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| #A [[generator matrix]] for the binary Golay code is '''I A''', where '''I''' is the 12×12 identity matrix, and '''A''' is the complement of the [[adjacency matrix]] of the [[icosahedron]].
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| == Practical applications of Golay codes ==
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| === NASA deep space missions ===
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| The [[Voyager program|Voyager]] 1 and 2 spacecraft needed to transmit hundreds of color pictures of [[Jupiter]] and [[Saturn]] in their 1979, 1980, and 1981 fly-bys within a constrained telecommunications bandwidth.
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| * Color image transmission required three times the amount of data, so the Golay (24,12,8) code was used. <ref>http://www-math.ucdenver.edu/~wcherowi/courses/m7409/mariner9talk.pdf</ref>
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| * This Golay code is only triple-error correcting, but it could be transmitted at a much higher data rate than the [[Hadamard code]] that was used during the Mariner mission.
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| === Radio communications ===
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| The new American government standards for [[automatic link establishment]] in [[high frequency]] radio systems specify the use of an extended (24,12) Golay block code for [[forward error correction]] (FEC).
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| * The Extended (24,12) Golay Code specified is a (24,12) block code.
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| * This code encodes 12 data bits to produce 24-bit code words.
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| * It is furthermore a systematic code, meaning that the 12 data bits are present in unchanged form in the code word.
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| The minimum [[Hamming distance]] between any two code words (the number of bits by which any pair of code words differs) is eight.
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| == See also ==
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| * [[Leech lattice]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{cite journal |author=Golay, Marcel J. E. |title=Notes on Digital Coding |journal=[[Proc. IRE]] |volume=37 |pages=657 |year=1949}}
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| * {{cite journal |author=Curtis, R. T. |title=A new combinatorial approach to M<sub>24</sub> |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|volume=79 |year=1976 |pages=25–42 |doi=10.1017/S0305004100052075}}
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| * {{cite book |author=[[Robert Griess|Griess, Robert L.]] |title=Twelve Sporadic Groups |publisher=Springer |year=1998 |pages=167 |isbn=978-3-540-62778-4}}
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| * {{cite book |author=Thompson, Thomas M. |title=From Error Correcting Codes through Sphere Packings to Simple Groups |series=Carus Mathematical Monographs |volume=21 |publisher=Mathematical Association of America |year=1983 |isbn=978-0-88385-023-7}}
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| * {{cite book |author=Greferath, Marcus |editor-last=Proakis |editor-first=John G.|chapter= Golay Codes |title=Encyclopedia of Telecommunications |publisher=Wiley |year=2003 |doi=10.1002/0471219282}}
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| {{CCSDS}}
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| [[Category:Error detection and correction]]
| |
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