Low-density parity-check code: Difference between revisions

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{{Use mdy dates|date = March 2019}}

'''Low-density parity-check (LDPC)''' codes are a class of [[error correction code]]s which (together with the closely- related [[turbo code]]s) have gained prominence in [[coding theory]] and [[information theory]] since the late 1990s. The codes today are widely used in applications ranging from wireless communications to flash-memory storage. Together with turbo codes, they sparked a revolution in coding theory, achieving order-of-magnitude improvements in performance compared to traditional error correction codes.<ref>{{Cite web |title=Turbo Codes Explained: History, Examples, and Applications - IEEE Spectrum |url=https://spectrum.ieee.org/turbo-codes |access-date=2024-12-18 |website=spectrum.ieee.org |language=en}}</ref>

Central to the performance of LDPC codes is their adaptability to the iterative [[belief propagation]] decoding algorithm. Under this algorithm, they can be designed to approach theoretical limits ([[Channel capacity|capacities]]) of many channels<ref>{{Cite journal |title=Design of capacity-approaching irregular low-density parity-check codes |url=https://ieeexplore.ieee.org/document/910578 |archive-url=http://web.archive.org/web/20240909161749/https://ieeexplore.ieee.org/document/910578/ |archive-date=2024-09-09 |access-date=2024-12-18 |journal=IEEE Transactions on Information Theory |date=2001 |doi=10.1109/18.910578 |language=en-US |last1=Richardson |first1=T.J. |last2=Shokrollahi |first2=M.A. |last3=Urbanke |first3=R.L. |volume=47 |issue=2 |pages=619–637 }}</ref> at low computation costs.

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LDPC codes were originally conceived by [[Robert G. Gallager]] (and are thus also known as Gallager codes). Gallager devised the codes in his doctoral dissertation<ref>{{cite thesis |last=Gallager |first=Robert G. |date= 1960 |title=Low density parity check codes |url=https://dspace.mit.edu/bitstream/handle/1721.1/11804/32786367-MIT.pdf |degree=Ph.D |publisher=Massachusetts Institute of Technology }}</ref> at the [[Massachusetts Institute of Technology]] in 1960.<ref>{{Cite news |last=Hardesty |first=L. |date=January 21, 2010 |title=Explained: Gallager codes |url=http://web.mit.edu/newsoffice/2010/gallager-codes-0121.html |access-date=August 7, 2013 |journal=MIT News}}</ref><ref name="G1962">{{cite journal |last=Gallager |first=R.G. |date=January 1962 |title=Low density parity check codes |journal=IRE Trans. Inf. Theory |volume=8 |issue=1 |pages=21–28 |doi=10.1109/TIT.1962.1057683 |s2cid=260490814 |hdl=1721.1/11804/32786367-MIT }}</ref> The codes were largely ignored at the time, as their iterative decoding algorithm (despite having linear complexity), was prohibitively computationally expensive for the hardware available.

Renewed interest in the codes emerged following the invention of the closely- related [[turbo code]]s (1993), whose similarly iterative decoding algorithm outperformed other codes used at that time. LDPC codes were subsequently rediscovered in 1996.<ref name="MacKay96">{{cite journal

|title=Near Shannon limit performance of low density parity check codes

|last1=MacKay |first1=David JC |author1-link=David J. C. MacKay|last2=Neal |first2=Radford M |author2-link=Radford M. Neal