|
|
| Line 1: |
Line 1: |
| 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]]
| |
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.
My page cheap psychic readings