Tanner graph

A Tanner graph is a bipartite graph used to specify 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 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 $R$ be the rate of the resulting linear code, let the degree of the digit nodes be $m$ and the degree of the subcode nodes be $n$. Then the rate of the code is bounded by

: $R\; geq\; 1\; -\; (1\; -\; n)m$

Computational complexity of Tanner graph based methods

The advantage of these recursive techniques is that they are computationally tractable. The codingalgorithm for Tanner graphs is extremely efficient in practice, although it is notguaranteed to converge except for cycle-free graphs, which are known not to admit asymptoticallygood codes. [T. Etzion, A. Trachtenberg, and A. Vardy, Which Codes have Cycle-Free Tanner Graphs?,IEEE Trans. Inf. Theory, 45:6.]

Notes

External links

* [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]