Low-density parity-check code

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A low-density parity-check code (LDPC code) is an error correcting code, a method of transmitting a message over a noisy transmission channel. While LDPC and other error correcting codes cannot guarantee perfect transmission, the probability of lost information can be made as small as desired. LDPC was the first code to allow data transmission rates close to the theoretical maximum, the Shannon Limit. Impractical to implement when developed in 1963, LDPC was forgotten. The next 30 or so years of information theory failed to produce anything one-third as effective and LDPC remains, in theory, the most effective developed to date (2006).

The explosive growth in information technology has produced a corresponding increase of commercial interest in the development of highly efficient data transmission codes as such codes impact everything from signal quality to battery life. Although implementation of LDPC codes has lagged that of other codes, notably the turbo code, the absence of encumbering software patents has made LDPC attractive to some and LDPC codes are positioned to become a standard in the developing market for highly efficient data transmission methods. In 2003 an LDPC code beat six turbo codes to become the new standard for the satellite transmission of digital television.

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Function

LDPC uses a sparse parity-check matrix. This sparse matrix is randomly generated, subject to the sparsity constraints. Decoding them is an NP-complete problem, but there are good approximate decoders. These codes were first designed by Gallager in 1962. See sparse graph code.

Below is a graph fragment of an example LDPC code using [[Forney's factor graph notation]]. A message is encoded by placing bits on the T's at the top such that the graphical constraints are satisfied. Specifically, all lines connecting to an <math>=</math> box have the same value, and all values connecting to a <math>+</math> box must sum to zero modulo two (i.e. must sum to an even number). Image:Ldpc code fragment factor graph.png If we ignore constraints for lines going out of the picture, then there are 8 possible 6 bit strings which correspond to valid codewords: (i.e., 000000, 011001, 110010, 111100, 101011, 100101, 001110, 010111). Thus this LDPC code fragment represents a 3-bit message with 6 bits. The purpose of this redundancy is to aid in recovering from channel errors.

Imagine that the 5th message, 101011, is transmitted across a channel and received with the 1st and 4th bit erased to yield *01*11. We know that the transmitted message must have satisfied the code constraints which we can represent by writing the received message on the top of the factor graph as shown below. Image:Ldpc code fragment factor graph w erasures.png We can now solve for the missing bits using an algorithm which is commonly referred to as belief propagation. In this case, the first step of belief propagation is to realize that the 4th bit must be 0 to satisfy the middle constraint. Image:Ldpc code fragment factor graph w erasures decode step 1.png Now that we have decoded the 4th bit, we realize that the 1st bit must be a 1 to satisfy the leftmost constraint. Image:Ldpc code fragment factor graph w erasures decode step 2.png Thus we are able to iteratively decode the message encoded with our LDPC code.

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References

David J.C. MacKay (2003) Information theory, inference and learning algorithms, CUP, ISBN 0521642981, (also available online)
Todd K. Moon (2005) Error Correction Coding, Mathematical Methods and Algorithms. Wiley, ISBN 0-471-64800-0 (Includes code)

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