Reed-Muller Codes Polarize

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.