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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.
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| == Origins ==
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| 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.
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| 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.
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| == Tanner graphs for linear block codes ==
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| [[Image:Tanner graph example.PNG|right|350px|Tanner graph with subcode and digit nodes]]
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| 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.
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| == Bounds proved by Tanner ==
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| Tanner proved the following bounds
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| 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
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| : <math> R \geq 1 - (1 - r)m \, </math>
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| == Computational complexity of Tanner graph based methods ==
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| The advantage of these recursive techniques is that they are computationally tractable. The coding
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| algorithm for Tanner graphs is extremely efficient in practice, although it is not
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| guaranteed to converge except for cycle-free graphs, which are known not to admit asymptotically
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| good codes.<ref>T. Etzion, A. Trachtenberg, and A. Vardy, Which Codes have Cycle-Free Tanner Graphs?, IEEE Trans. Inf. Theory, 45:6.</ref>
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| == Applications of Tanner graph ==
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| [[Zemor's decoding algorithm]], which is a recursive low-complexity approach to code construction, is based on Tanner graphs.
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| == Notes ==
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| <references/>
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| *[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1056404 Michael Tanner's Original paper]
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| *[http://www.uic.edu/index.html/admin_tanner.shtml Michael Tanner's page]
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| [[Category:Coding theory]]
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| [[Category:Application-specific graphs]]
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