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{{infobox code | |||
| name = Extended binary Golay code | |||
| image = [[File:BinaryGolayCode.svg|250px]] | |||
| image_caption = Its 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 [[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]]. | |||
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. | |||
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. | |||
== Mathematical definition == | |||
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. | |||
* 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. | |||
* Up to relabeling coordinates, ''W'' is unique. | |||
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. | |||
The [[automorphism group]] of the binary Golay code is the [[Mathieu group]] <math>M_{23}</math>. | |||
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''. | |||
The Golay [[code word]]s of weight eight are elements of the S(5,8,24) [[Steiner system]]. | |||
== Constructions == | |||
#[[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>. | |||
#[[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 ∞.) | |||
#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> | |||
#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. | |||
#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. | |||
#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]]. | |||
== Practical applications of Golay codes == | |||
=== NASA deep space missions === | |||
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. | |||
* 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> | |||
* 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. | |||
=== Radio communications === | |||
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). | |||
* The Extended (24,12) Golay Code specified is a (24,12) block code. | |||
* This code encodes 12 data bits to produce 24-bit code words. | |||
* It is furthermore a systematic code, meaning that the 12 data bits are present in unchanged form in the code word. | |||
The minimum [[Hamming distance]] between any two code words (the number of bits by which any pair of code words differs) is eight. | |||
== See also == | |||
* [[Leech lattice]] | |||
==Notes== | |||
{{Reflist}} | |||
==References== | |||
* {{cite journal |author=Golay, Marcel J. E. |title=Notes on Digital Coding |journal=[[Proc. IRE]] |volume=37 |pages=657 |year=1949}} | |||
* {{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}} | |||
* {{cite book |author=[[Robert Griess|Griess, Robert L.]] |title=Twelve Sporadic Groups |publisher=Springer |year=1998 |pages=167 |isbn=978-3-540-62778-4}} | |||
* {{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}} | |||
* {{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}} | |||
{{CCSDS}} | |||
[[Category:Error detection and correction]] | |||
Revision as of 16:12, 14 September 2013
Template:Infobox code Template:Infobox 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. These codes are named in honor of Marcel J. E. Golay.
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. 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 of the code, and the minimum Hamming distance between two codewords, respectively.
Mathematical definition
In mathematical terms, the extended binary Golay code consists of a 12-dimensional subspace W of the space V=F224 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.
- 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 weights of 0, 8, 12, 16, or 24.
- Up to relabeling coordinates, W is unique.
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.
The automorphism group of the binary Golay code is the Mathieu group . The automorphism group of the extended binary Golay code is the Mathieu group . The other Mathieu groups occur as stabilizers of one or several elements of W.
The Golay code words of weight eight are elements of the S(5,8,24) Steiner system.
Constructions
- 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 w1 = 0, define w2, w3, ..., w12 by the rule that wn 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 w1, ..., w12.
- Quadratic residue code: Consider the set N of quadratic non-residues (mod 23). This is an 11-element subset of the cyclic group Z/23Z. Consider the translates t+N of this subset. Augment each translate to a 12-element set St 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 St together with the word consisting of all basis vectors. (The perfect code is obtained by leaving out ∞.)
- As a Cyclic code: The perfect G23 code can be constructed via factorization of , it is the code generated by
- From the 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 differences 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 into 4 planes. [1] Nowadays often the compact approach of Conway's hexacode, that uses a 4×6 array of square cells, is used.
- 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.
- 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.
Practical applications of Golay codes
NASA deep space missions
The 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.
- Color image transmission required three times the amount of data, so the Golay (24,12,8) code was used. [2]
- 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.
Radio communications
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).
- The Extended (24,12) Golay Code specified is a (24,12) block code.
- This code encodes 12 data bits to produce 24-bit code words.
- It is furthermore a systematic code, meaning that the 12 data bits are present in unchanged form in the code word.
The minimum Hamming distance between any two code words (the number of bits by which any pair of code words differs) is eight.
See also
Notes
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References
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534