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Reed-MullerÌý(RM)ÌýcodesÌýwere 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 polarÌýcodes. RMÌýcodesÌýand polarÌýcodesÌýare generated by the same matrixÌýG_m=Ìý[1Ìý&Ìý0Ìý\\Ìý1Ìý&Ìý1]{\otimesÌým}G_m=Ìý[1Ìý&Ìý0Ìý\\Ìý1Ìý&Ìý1]{\otimesÌým}Ìýbut using different subset of rows.Ìý RMÌýcodesÌýselect simply rows having largest weights. PolarÌýcodesÌýselect instead rows having the largest conditional mutual information proceeding top to down inÌýGmGm; 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. RMÌýcodesÌýare yet to be proved to have such a property, despite the recent success for the erasure channel.Ìý

In this talk, we connect RMÌýcodesÌýto polarization theory. We show that proceeding in the RM code order, i.e., not top-to-down but from the lightest to the heaviest rows inÌýGmGm, the conditional mutual information again polarizes. We further demonstrate that it does so faster than for polarÌýcodes. This implies thatÌýGmGmÌýcontains 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 RMÌýcodesÌýto 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 twoÌýcodesÌýare at least similar and give further evidence that they are indeed the same.Ìý

This talk is based on joint work with Emmanuel Abbe. The paper will appear at FOCS this year.

Min Ye received his B.S. in Electrical Engineering from Peking University, Beijing, China in 2012, and his Ph.D. in the Department of Electrical and Computer Engineering, University of Maryland, College Park in 2017. He is currently a postdoctoral researcher at Princeton University.ÌýHe is the recipient of the IEEE Data Storage Best Paper Award. His research interests include coding theory, information theory, differential privacy, and machine learning.