Reed–Muller code: Difference between revisions

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'''Reed–Muller codes''' are [[error-correcting code]]s that are used in wireless communications applications, particularly in deep-space communication<ref>{{Citation|last=Massey|first=James L.|titledate=Deep1992|pages=1–17|publisher=Springer-space communications and coding: A marriage made in heavenVerlag|datelanguage=1992en|urldoi=https://link.springer.com/chapter/10.1007/BFb0036046bfb0036046|workisbn=978-3540558514|title=Advanced Methods for Satellite and Deep Space Communications|pagesvolume=1–17182|publisherseries=Springer-VerlagLecture Notes in Control and Information Sciences|languagechapter=enDeep-space communications and coding: A marriage made in heaven|doiciteseerx=10.1007/bfb0036046|isbn=3540558519|access-date=2018-08-311.1.36.4265}}[[https://www.isiweb.ee.ethz.ch/archive/massey_pub/pdf/BI321.pdf pdf]]</ref>. Moreover, the proposed [[5G|5G standard]]<ref>{{cite web|url=http://www.3gpp.org/ftp/tsg_ran/WG1_RL1/TSGR1_87/Report/Final_Minutes_report_RAN1%2387_v100.zip|title=3GPP RAN1 meeting #87 final report|publisher=3GPP|accessdate=31 August 2017}}</ref> relies on the closely related [[Polar code (coding theory)|polar codes]]<ref>{{Cite webjournal|url=https://ieeexplore.ieee.org/abstract/document/5075875/|title=Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels - IEEE Journals & Magazine|websitejournal=ieeexplore.ieee.orgIEEE Transactions on Information Theory|volume=55|issue=7|pages=3051–3073|language=en-US|access-date=2018-08-31|doi=10.1109/TIT.2009.2021379|year=2009|last1=Arikan|first1=Erdal|hdl=11693/11695}}</ref> for error correction in the control channel. Due to their favorable theoretical and mathematical properties, Reed–Muller codes have also been extensively studied in [[theoretical computer science]].

Reed–Muller codes generalize the [[Reed–Solomon error correction|Reed–Solomon codes]] and the [[Hadamard code|Walsh–Hadamard code]]. Reed–Muller codes are [[Linear code|linear block codes]] that are [[locally testable code|locally testable]], [[locally decodable code|locally decodable]], and [[List decoding|list decodable]]. These properties make them particularly useful in the design of [[probabilistically checkable proof]]s.

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== Further reading ==

* {{cite book | author=Shu Lin |author2=Daniel Costello | title=Error Control Coding | edition=2 | year=2005 | publisher=Pearson | isbn=978-0-13-017973-69 }} Chapter 4.

* {{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2 | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=86 | year=1992 | isbn=978-3-540-54894-72 }} Chapter 4.5.

==External links==

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* [http://octave.sourceforge.net/communications/function/reedmullergen.html GPL Matlab-implementation of RM-codes ]

* [http://octave.svn.sourceforge.net/viewvc/octave/trunk/octave-forge/main/comm/inst/reedmullergen.m?revision=9852&view=markup Source GPL Matlab-implementation of RM-codes]

*{{cite journal |last1=Weiss |first1=E. |title=Generalized Reed-Muller codes |journal=Information and Control |date=September 1962 |volume=5 |issue=3 |pages=213–222 |doi=10.1016/s0019-9958(62)90555-7 |url=https://doi.org/10.1016/S0019-9958(62)90555-7 |issn=0019-9958}}

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