Reed-Muller Codes Polarize

Emmanuel Abbe, Min Ye

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Scopus citations

Abstract

Reed-Muller (RM) codes were introduced in 1954 and have long been conjectured to achieve Shannon's capacity on symmetric channels. The activity on this conjecture has recently been revived with the emergence of polar codes. RM codes and polar codes are generated by the same matrix G-m= [1/1 0/1] ⊗m but using different subset of rows. RM codes select simply rows having largest weights. Polar codes select instead rows having the largest conditional mutual information proceeding top to down in G-m; while this is a more elaborate and channel-dependent rule, the top-To-down ordering allows Arikan to show that the conditional mutual information polarizes, and this gives directly a capacity-Achieving code on any symmetric channel. RM codes are yet to be proved to have such a property, despite the recent success for the erasure channel. In this paper, we connect RM codes to polarization theory. We show that proceeding in the RM code ordering, i.e., not top-To-down but from the lightest to the heaviest rows in G-m, the conditional mutual information again polarizes. We further demonstrate that it does so faster than for polar codes. This implies that G-m contains another code, different than the polar code and called here the twin-RM code, that is provably capacity-Achieving on any symmetric channel. This gives in particular a necessary condition for RM codes to achieve capacity on symmetric channels. It further gives a sufficient condition if the rows with largest conditional mutual information correspond to the heaviest rows, i.e., if the twin-RM code is the RM code. We demonstrate here that the two codes are at least similar and give further evidence that they are indeed the same.

Original languageEnglish (US)
Title of host publicationProceedings - 2019 IEEE 60th Annual Symposium on Foundations of Computer Science, FOCS 2019
PublisherIEEE Computer Society
Pages273-286
Number of pages14
ISBN (Electronic)9781728149523
DOIs
StatePublished - Nov 2019
Event60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019 - Baltimore, United States
Duration: Nov 9 2019Nov 12 2019

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2019-November
ISSN (Print)0272-5428

Conference

Conference60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019
CountryUnited States
CityBaltimore
Period11/9/1911/12/19

All Science Journal Classification (ASJC) codes

  • Computer Science(all)

Keywords

  • Reed-Muller codes
  • Shannon theory
  • capacity-Achieving codes
  • polar codes

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