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.|title=Deep-space communications and coding: A marriage made in heaven|date=1992|url=https://link.springer.com/chapter/10.1007/BFb0036046|work=Advanced Methods for Satellite and Deep Space Communications|pages=1–17|publisher=Springer-Verlag|language=en|doi=10.1007/bfb0036046|isbn=3540558519|access-date=2018-08-31}}[[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 web|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|website=ieeexplore.ieee.org|language=en-US|access-date=2018-08-31}}</ref> for itserror channelcorrection codingin needsthe 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.