en>Omnipaedista: per MOS:BOLDSYN

2013-07-16T20:37:50Z

← Older revision		Revision as of 22:37, 16 July 2013
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	~~Alyson~~ is ~~what my spouse enjoys to contact me but I don~~'~~t like when people use my complete name. Playing badminton~~ is a ~~factor that he~~ is ~~totally addicted~~ to. ~~North Carolina~~ is exactly ~~where we~~'~~ve been living for many years~~ and ~~will never transfer~~. ~~Invoicing~~ is ~~what I do~~.<br><br>~~Feel free to visit my page :: free psychic readings~~ ([http://~~brazil~~.~~amor~~-~~amore~~.~~com/irboothe mouse click~~ the ~~following web page~~])		{{infobox code
			\| name = Perfect ternary Golay code
			\| image =
			\| image_caption =
			\| namesake = [[Marcel J. E. Golay]]
			\| type = [[Linear block code]]
			\| block_length = 11
			\| message_length = 6
			\| rate = 6/11 ~ 0.545
			\| distance = 5
			\| alphabet_size = 3
			\| notation = <math>[11,6,5]_3</math>-code
			}}
			{{infobox code
			\| name = Extended ternary Golay code
			\| image =
			\| image_caption =
			\| namesake = [[Marcel J. E. Golay]]
			\| type = [[Linear block code]]
			\| block_length = 12
			\| message_length = 6
			\| rate = 6/12 = 0.5
			\| distance = 6
			\| alphabet_size = 3
			\| notation = <math>[12,6,6]_3</math>-code
			}}

			In [[coding theory]], the '''ternary Golay codes''' are two closely related [[error-correcting code]]s.
			The code generally known simply as the '''ternary Golay code''' is an <math>[11, 6, 5]_3</math>-code, that is, it is a [[linear code]] over a [[ternary numeral system\|ternary]] alphabet; the [[Block_code#The_distance_d\|relative distance]] of the code is as large as it possibly can be for a ternary code, and hence, the ternary Golay code is a [[perfect code]].
			The '''extended ternary Golay code''' is a [12, 6, 6] [[linear code]] obtained by adding a zero-sum [[check digit]] to the [11, 6, 5] code.
			In finite [[group theory]], the extended ternary Golay code is sometimes referred to as the ternary Golay code.{{citation missing\|date=February 2013}}

			== Properties ==

			=== Ternary Golay code ===
			The ternary Golay code consists of 3<sup>6</sup> = 729 codewords.
			Its [[parity check matrix]] is
			:<math>

			\left[
			\begin{array}{ccccccccccc}

			1 & 1 & 1 & 2 & 2 & 0 & 1 & 0 & 0 & 0 & 0\\

			1 & 1 & 2 & 1 & 0 & 2 & 0 & 1 & 0 & 0 & 0\\

			1 & 2 & 1 & 0 & 1 & 2 & 0 & 0 & 1 & 0 & 0\\

			1 & 2 & 0 & 1 & 2 & 1 & 0 & 0 & 0 & 1 & 0\\

			1 & 0 & 2 & 2 & 1 & 1 & 0 & 0 & 0 & 0 & 1

			\end{array}
			\right].</math>
			Any two different codewords differ in at least 5 positions.
			Every ternary word of length 11 has a [[Hamming distance]] of at most 2 from exactly one codeword.
			The code can also be constructed as the [[quadratic residue code]] of length 11 over the [[finite field]] '''F'''<sub>3</sub>.

			Used in a [[football pool]] with 11 games, the ternary Golay code corresponds to 729 bets and guarantees exactly one bet with at most 2 wrong outcomes.

			The set of codewords with Hamming weight 5 is a 3-(11,5,4) [[t-design\|design]].

			=== Extended ternary Golay code ===
			The [[enumerator polynomial\|complete weight enumerator]] of the extended ternary Golay code is
			:<math>x^{12}+y^{12}+z^{12}+22\left(x^6y^6+y^6z^6+z^6x^6\right)+220\left(x^6y^3z^3+y^6z^3x^3+z^6x^3y^3\right).</math>

			The [[automorphism group]] of the extended ternary Golay code is 2.''M''<sub>12</sub>, where ''M''<sub>12</sub> is the [[Mathieu group M12]].

			The extended ternary Golay code can be constructed as the span of the rows of a [[Hadamard matrix]] of order 12 over the field '''F'''<sub>3</sub>.

			Consider all codewords of the extended code which have just six nonzero digits. The sets of positions at which these nonzero digits occur form the [[Steiner system]] S(5, 6, 12).

			== History ==

			The ternary Golay code was first introduced by the [[Finland\|Finnish]] football pool enthusiast [[Juhani Virtakallio]], who published it in 1947 in issues 27, 28 and 33 of the football [[magazine]] ''[[Veikkaaja]]''. {{harv\|Barg\|1993\|loc=p.25}}
			It was independently rediscovered by {{harvs\|txt\|authorlink=Marcel J. E. Golay\|last=Golay\|year=1949}}.

			== See also ==

			* [[Binary Golay code]]

			==References==

			*{{Citation \| last1=Barg \| first1=Alexander \| title=At the dawn of the theory of codes \| url=http://dx.doi.org/10.1007/BF03025254 \| doi=10.1007/BF03025254 \| id={{MR\|1199273}} \| year=1993 \| journal=[[The Mathematical Intelligencer]] \| issn=0343-6993 \| volume=15 \| issue=1 \| pages=20–26}}
			* M.J.E. Golay, Notes on digital coding, ''Proceedings of the [[Institute of Radio Engineers\|I.R.E.]]'' 37 (1949) 657
			* I.F. Blake (ed.), ''Algebraic Coding Theory: History and Development'', Dowden, Hutchinson & Ross, Stroudsburg 1973
			* [[J. H. Conway]] and [[N. J. A. Sloane]], ''[[Sphere packing\|Sphere Packings]], [[Lattice (group)\|Lattices]] and [[Group (mathematics)\|Groups]]'', [[Springer Science+Business Media\|Springer]], New York, Berlin, Heidelberg, 1988.
			* [[Robert L. Griess]], ''Twelve [[Sporadic group\|Sporadic Groups]]'', Springer, 1998.
			* G. Cohen, I. Honkala, S. Litsyn, A. Lobstein, ''Covering Codes'', [[Elsevier]] (1997) ISBN 0-444-82511-8
			* Th. M. Thompson, ''From Error Correcting Codes through Sphere Packings to [[Simple_group\|Simple Groups]]'', [[The Mathematical Association of America]] 1983, ISBN 0-88385-037-0

			[[Category:Coding theory]]
			[[Category:Finite fields]]

			[[ja:3元ゴレイ符号]]

en>Michael Hardy: /* Mathematical foundation */

2010-06-01T13:18:54Z

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Alyson is what my spouse enjoys to contact me but I don't like when people use my complete name. Playing badminton is a factor that he is totally addicted to. North Carolina is exactly where we've been living for many years and will never transfer. Invoicing is what I do.<br><br>Feel free to visit my page :: free psychic readings ([http://brazil.amor-amore.com/irboothe mouse click the following web page])

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en>Michael Hardy: /* Mathematical foundation */