Abstract
These days we encounter many digital storage and communication devices in our daily lives. They contain error correcting codes that operate when data is read from storage devices or received via communication devices. For example, you can listen to music on a compact disc even if its surface is scratched. This article introduces low density parity check (LDPC) codes and the sum-product decoding algorithm. LDPC codes, one class of error correcting codes, have been used for practical applications such as hard disk drives and satellite digital broadcast systems because their performance closely approaches the theoretical limit with manageable computational complexity. In particular, it is shown that an optimal decoding algorithm from the viewpoint of probabilistic inference can be derived with LDPC codes.
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Notes
- 1.
Sometimes called symbol MAP decoding to distinguish it from the MAP algorithm that maximizes the APP of the codeword \({\mathbf X}\).
- 2.
More precisely, it is the opposite way to marginalization. Marginalization is the computation to obtain the probability distribution with fewer variables from a multivariate probability distribution, e.g. the marginal distribution of the random variable \(A\) is given by \(\displaystyle P_{A}(a) = \sum _{b\in {\fancyscript{B}}} P_{AB}(a,b)\), where \({\fancyscript{B}}\) is the domain of the random variable \(B\).
References
R.G. Gallager, in Low-Density Parity-Check Codes (MIT Press, Cambridge, 1963)
S. Lin, D.J. Costello, in Error Control Coding, 2nd edn. (Prentice Hall, Englewood Cliffs, 2004)
T.J. Richardson, R. Urbanke, in Modern Coding Theory (Cambridge University Press, Cambridge, 2008)
W.E. Ryan, S. Lin, in Channel Codes: Classical and Modern (Cambridge University Press, Cambridge, 2009)
C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. (1948)
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© 2014 Springer Japan
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Uchikawa, H. (2014). Error Correcting Codes Based on Probabilistic Decoding and Sparse Matrices. In: Nishii, R., et al. A Mathematical Approach to Research Problems of Science and Technology. Mathematics for Industry, vol 5. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55060-0_36
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DOI: https://doi.org/10.1007/978-4-431-55060-0_36
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