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 $\begin{aligned} G_{m}= \left[{\begin{smallmatrix}1 & 0 1 & 1 \end{smallmatrix}}\right]^{\otimes m} \end{aligned}$ 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 article, 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. Here “polarization” means that almost all the conditional mutual information becomes either very close to 0 or very close to 1. Polarization itself is a necessary condition for RM codes to achieve capacity on symmetric channels while polarization together with a strong order on the conditional mutual information gives a sufficient condition, where strong order means that rows with larger weight always correspond to larger conditional mutual information. Although we are not able to prove the strong order, we establish a partial order on the conditional mutual information, which is a subset of the strong order. While the main results of this article–polarization together with the partial order–provide some advances on the capacity-achieving conjecture of RM codes, we emphasize that our results do not allow us to prove the conjecture.