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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. | |||
== 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]] | |||
Revision as of 22:11, 4 October 2013
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[1] as a means to create larger error correcting codes from smaller ones using recursive techniques. He generalized the 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
Tanner graphs are 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 be the rate of the resulting linear code, let the degree of the digit nodes be and the degree of the subcode nodes be . 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
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.[2]
Applications of Tanner graph
Zemor's decoding algorithm, which is a recursive low-complexity approach to code construction, is based on Tanner graphs.
Notes
- ↑ 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
- ↑ T. Etzion, A. Trachtenberg, and A. Vardy, Which Codes have Cycle-Free Tanner Graphs?, IEEE Trans. Inf. Theory, 45:6.