Low-density parity-check code: Difference between revisions

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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">[[David J.C. MacKay]] and Radford M. Neal, "Near Shannon Limit Performance of Low Density Parity Check Codes," Electronics Letters, July 1996</ref> Initial industry preference for LDPC codes over turbo codes stemmed from patent-related constraints on the latter<ref name="Closing">{{cite journal |author=Erico Guizzo |date=Mar 1, 2004 |title=CLOSING IN ON THE PERFECT CODE |url=https://spectrum.ieee.org/closing-in-on-the-perfect-code |url-status=dead |journal=IEEE Spectrum |archive-url=https://web.archive.org/web/20210902170851/https://spectrum.ieee.org/closing-in-on-the-perfect-code |archive-date=September 2, 2021}} "Another advantage, perhaps the biggest of all, is that the LDPC patents have expired, so companies can use them without having to pay for intellectual-property rights."</ref>. Over the time that has elapsed since their discovery, advances in LDPC codes have seen them surpass turbo codes in terms of [[error floor]] and performance in the higher [[code rate]] range, leaving turbo codes better suited for the lower code rates only.<ref>[http://deepspace.jpl.nasa.gov/dsndocs/810-005/208/208B.pdf Telemetry Data Decoding, Design Handbook]</ref> Although the fundamental patent for turbo codes has expired (on August 29, 2013),<ref>{{cite patent|country=US|number=5446747|url=https://www.google.com/patents/US5446747}}</ref><ref>{{cite journal |last=Mackenzie |first=D. |date=9 July 2005 |title=Communication speed nears terminal velocity |journal=New Scientist}}</ref> LDPC codes are now still being preferred for their technical merits.

Theoretical interest in LDPC codes also follows from their amenability to theoreticalmathematical analysis. In his dissertation, Gallager showed that LDPC codes achieve the [[Gilbert–Varshamov bound for linear codes]] over binary fields with high probability. Over the [[binary erasure channel]], code sequences were designed at rates arbitrary close to channel capacity, with provably vanishing decoding error probability and linear decoding complexity<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=2024-12-19 |website=ieeexplore.ieee.org |language=en-US}}</ref>. In 2020 it was shown that Gallager's LDPC codes achieve [[list decoding]] capacity and also achieve the [[Gilbert–Varshamov bound for linear codes]] over general fields.<ref name="MRRSW20">{{cite journal |last1=Mosheiff |first1=J. |last2=Resch |first2=N. |last3=Ron-Zewi |first3=N. |last4=Silas |first4=S. |last5=Wootters |first5=M. |date=2020 |title=Low-Density Parity-Check Codes Achieve List-Decoding Capacity |journal=SIAM Journal on Computing |volume=53 |issue=FOCS 2020 |pages=38–73 |arxiv=1909.06430 |doi=10.1137/20M1365934 |s2cid=244549036}}</ref>

==Applications==