Rational normal curve - Revision history

en>Michael Hardy: \ldots between commas; \cdots between colons

2014-11-04T17:27:26Z

\ldots between commas; \cdots between colons

← Older revision		Revision as of 19:27, 4 November 2014
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99.241.166.168: plain html to {{math| ...}}

2014-02-20T00:27:47Z

plain html to {{math| ...}}

← Older revision		Revision as of 02:27, 20 February 2014
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	~~In [[coding theory]], a~~ '~~''Tanner graph''', named after Michael Tanner,~~ is a [[bipartite graph]] used to state constraints or equations which specify [[error correcting codes]]. In [[coding theory]], Tanner graphs are used to construct longer codes from smaller ones. Both encoders and decoders employ these graphs extensively.		The author's title is Christy Brookins. I am presently a travel agent. North Carolina is the location he loves most but now he is considering other options. I am truly fond of handwriting but I can't make it my profession really.<br><br>My page [http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 cheap psychic readings]

	~~== Origins ==~~
	Tanner graphs were proposed by Michael Tanner<ref>[http://www.copyright.gov/disted/comments/init040.pdf R. Michael Tanner Professor of Computer Science, School of Engineering University of California, Santa Cruz Testimony before Representatives of the United States Copyright Office February 10, 1999]</ref> as a means to create larger error correcting codes from smaller ones using recursive techniques. He generalized the [[Peter Elias\|techniques of Elias]] for product codes.

	~~Tanner discussed lower bounds on the codes obtained from these graphs irrespective of the specific characteristics of the codes which were being used to construct larger codes.~~

	~~== Tanner graphs for linear block codes ==~~
	~~[[Image:Tanner graph example.PNG\|right\|350px\|Tanner graph with subcode and digit nodes]]~~
	~~Tanner graphs are [[bipartite graph\|partitioned]] into subcode nodes and digit nodes. For linear block codes, the subcode nodes denote rows of the [[parity-check matrix]] H~~. ~~The digit nodes represent the columns of the matrix H. An edge connects~~ a ~~subcode node to a digit node if a nonzero entry exists in the intersection of the corresponding row and column.~~

	~~== Bounds proved by Tanner ==~~
	~~Tanner proved the following bounds~~

	~~Let <math> R </math> be the rate of the resulting linear code, let the degree of the digit nodes be <math> m </math> and the degree of the subcode nodes be <math> n </math>~~. ~~If each subcode node~~ is ~~associated with a linear code (n,k) with rate r = k/n, then the rate of~~ the ~~code~~ is ~~bounded by~~

	~~: <math> R \geq 1 - (1 - r)m \, </math>~~

	~~== Computational complexity~~ of ~~Tanner graph based methods ==~~
	~~The advantage of these recursive techniques is that they are computationally tractable. The coding~~
	~~algorithm for Tanner graphs is extremely efficient in practice, although~~ it ~~is not~~
	~~guaranteed to converge except for cycle-free graphs, which are known not to admit asymptotically~~
	~~good codes.<ref>T. Etzion, A. Trachtenberg, and A. Vardy, Which Codes have Cycle-Free Tanner Graphs?, IEEE Trans. Inf. Theory, 45:6~~.<~~/ref~~>

	~~== Applications of Tanner graph ==~~
	~~[[Zemor's decoding algorithm]], which is a recursive low-complexity approach to code construction, is based on Tanner graphs.~~

	~~== Notes ==~~
	<~~references/~~>
	*[http://~~ieeexplore.ieee~~.~~org~~/~~xpls/abs_all~~.~~jsp~~?~~arnumber~~=~~1056404 Michael Tanner's Original paper]~~
	*[http://www.uic.edu/index.html/admin_tanner.shtml Michael Tanner's page]

	~~[[Category:Coding theory]]~~
	~~[[Category:Application-specific graphs]~~]

en>Danneks: /* Alternate parameterization */ wrong link

2013-10-04T20:11:50Z

Alternate parameterization: wrong link

← Older revision		Revision as of 22:11, 4 October 2013
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	~~Marvella~~ is ~~what you can call her but it's not~~ the ~~most feminine title out there~~. ~~Body building~~ is 1 of ~~the issues I love most~~. ~~Hiring has been my profession~~ for ~~some time but I've already utilized~~ for ~~an additional 1~~. ~~Her husband~~ and ~~her reside in Puerto Rico but she will~~ have ~~to transfer one working day or another~~.<br><br>~~Check out my weblog~~ ... ~~at home std testing (~~[http://www.~~pinaydiaries~~.~~com~~/~~blog~~/~~104745 click the next website~~ page])		In [[coding theory]], a '''Tanner graph''', named after Michael Tanner, is a [[bipartite graph]] used to state constraints or equations which specify [[error correcting codes]]. In [[coding theory]], Tanner graphs are used to construct longer codes from smaller ones. Both encoders and decoders employ these graphs extensively.

			== Origins ==
			Tanner graphs were proposed by Michael Tanner<ref>[http://www.copyright.gov/disted/comments/init040.pdf R. Michael Tanner Professor of Computer Science, School of Engineering University of California, Santa Cruz Testimony before Representatives of the United States Copyright Office February 10, 1999]</ref> as a means to create larger error correcting codes from smaller ones using recursive techniques. He generalized the [[Peter Elias\|techniques of Elias]] for product codes.

			Tanner discussed lower bounds on the codes obtained from these graphs irrespective of the specific characteristics of the codes which were being used to construct larger codes.

			== Tanner graphs for linear block codes ==
			[[Image:Tanner graph example.PNG\|right\|350px\|Tanner graph with subcode and digit nodes]]
			Tanner graphs are [[bipartite graph\|partitioned]] into subcode nodes and digit nodes. For linear block codes, the subcode nodes denote rows of the [[parity-check matrix]] H. The digit nodes represent the columns of the matrix H. An edge connects a subcode node to a digit node if a nonzero entry exists in the intersection of the corresponding row and column.

			== Bounds proved by Tanner ==
			Tanner proved the following bounds

			Let <math> R </math> be the rate of the resulting linear code, let the degree of the digit nodes be <math> m </math> and the degree of the subcode nodes be <math> n </math>. If each subcode node is associated with a linear code (n,k) with rate r = k/n, then the rate of the code is bounded by

			: <math> R \geq 1 - (1 - r)m \, </math>

			== Computational complexity of Tanner graph based methods ==
			The advantage of these recursive techniques is that they are computationally tractable. The coding
			algorithm for Tanner graphs is extremely efficient in practice, although it is not
			guaranteed to converge except for cycle-free graphs, which are known not to admit asymptotically
			good codes.<ref>T. Etzion, A. Trachtenberg, and A. Vardy, Which Codes have Cycle-Free Tanner Graphs?, IEEE Trans. Inf. Theory, 45:6.</ref>

			== Applications of Tanner graph ==
			[[Zemor's decoding algorithm]], which is a recursive low-complexity approach to code construction, is based on Tanner graphs.

			== Notes ==
			<references/>
			*[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1056404 Michael Tanner's Original paper]
			*[http://www.uic.edu/index.html/admin_tanner.shtml Michael Tanner's page]

			[[Category:Coding theory]]
			[[Category:Application-specific graphs]]

en>R.e.b.: Adding/removing wikilink(s)

2012-07-22T01:05:09Z

Adding/removing wikilink(s)

New page

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