Editing Binary Golay code (section)

{{Short description|Type of linear error-correcting code}}
{{infobox code
 | name           = Extended binary Golay code
 | image          = [[File:BinaryGolayCode.svg|250px]]
 | image_caption  = [[Generator matrix]]
 | namesake       = [[Marcel J. E. Golay]]
 | type           = [[Linear block code]]
 | block_length   = 24
 | message_length = 12
 | rate           = 12/24 = 0.5
 | distance       = 8
 | alphabet_size  = 2
 | notation       = <math>[24,12,8]_2</math>-code
}}
{{infobox code
 | name           = Perfect binary Golay code
 | image          =
 | image_caption  =
 | namesake       = [[Marcel J. E. Golay]]
 | type           = [[Linear block code]]
 | block_length   = 23
 | message_length = 12
 | rate           = 12/23 ~ 0.522
 | distance       = 7
 | alphabet_size  = 2
 | notation       = <math>[23,12,7]_2</math>-code
}}
In [[mathematics]] and [[electronics engineering]], a '''binary Golay code''' is a type of linear [[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.<ref>{{harvnb|Thompson|1983}}</ref> These codes are named in honor of [[Marcel J. E. Golay]] whose 1949 paper<ref>{{cite journal |last1=Golay |first1=Marcel J. E. |title=Notes on Digital Coding |journal=[[Proc. IRE]] |volume=37 |pages=657 |year=1949 |url=https://pierre-hyvernat.apps.math.cnrs.fr/data/Enseignement/2223/info602/TP-Golay/golay_paper.pdf |archiveurl=https://web.archive.org/web/20230410124137/https://www.lama.univ-savoie.fr/pagesmembres/hyvernat/Enseignement/2223/info602/TP-Golay/golay_paper.pdf |archivedate=April 10, 2023}}</ref> introducing them has been called, by [[E. R. Berlekamp]], the "best single published page" in [[coding theory]].<ref>{{citation|first=E. R.|last=Berlekamp|title=Key Papers in the Development of Coding Theory|year=1974|publisher=I.E.E.E. Press|page=4}}</ref>

{{anchor|Extended binary Golay code|Perfect binary Golay code}}There are two closely related binary Golay codes.  The '''extended binary Golay code''', ''G''<sub>24</sub> (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 4-bit errors can be detected.  
The other, the '''perfect binary Golay code''', ''G''<sub>23</sub>, 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 coding 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.