Low-density parity-check code: Difference between revisions

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LDPC codes are a class of [[Error correction code|error correction codes]] which (together with the closely-related [[Turbo code|turbo codes]]) have gained prominence in [[coding theory]] and [[information theory]] in 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 whichUnder this algorithm, they werehave been shown to approach theoretical limits ([[Channel capacity|capacities]]) of many channels<ref>{{Cite web |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 |website=ieeexplore.ieee.org |language=en-US}}</ref> at low computation costcosts. Theoretically, analysis of the codes focuses on sequences of codes of fixed [[code rate]] and increasing [[block length]]. When considering the decoding of codes in the sequence under belief propagation, in many casecases the decoding error can be proven to be vanishingly small (approaches zero with the block length), at a complexity that is linear in the block length.

This theoretical performance is made possible using a flexible design method that is based on sparse [[Tanner graph|Tanner graphs]] (specialized [[bipartite graph|bipartite graphs]]).<ref>{{citation |author=Amin Shokrollahi |url=http://www.ics.uci.edu/~welling/teaching/ICS279/LPCD.pdf |title=LDPC Codes: An Introduction |archive-url=https://web.archive.org/web/20170517034849/http://www.ics.uci.edu/~welling/teaching/ICS279/LPCD.pdf |archive-date=2017-05-17}}</ref>