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 Gm = [1110]⊗m but using different subset of rows. RM codes 1 1 select simply rows having largest weights. Polar codes select instead rows having the largest conditional mutual information proceeding top to down in Gm; while this is a more elaborate and channel-dependent rule, the top-to-down ordering allows Arıkan 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 Gm, 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.

中文翻译：

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里德-米勒码极化


Reed-Muller (RM) 码于 1954 年推出，长期以来一直被推测可以在对称信道上实现香农的容量。随着极地码的出现，这一猜想的活动最近又重新兴起。 RM 码和 Polar 码由相同的矩阵 Gm = [1110]⊗m 但使用不同的行子集生成。 RM 代码 1 1 仅选择具有最大权重的行。 Polar 码改为在 Gm 中从上到下选择具有最大条件互信息的行；虽然这是一个更复杂且依赖于信道的规则，但自上而下的排序允许 Arıkan 表明条件互信息极化，并且这直接给出了任何对称信道上的容量实现代码。尽管擦除通道最近取得了成功，但 RM 代码尚未被证明具有这样的属性。在本文中，我们将 RM 码与极化理论联系起来。我们证明，按照 RM 代码排序，即不是从上到下，而是从 Gm 中最轻的行到最重的行，条件互信息再次极化。这里的“极化”是指几乎所有的条件互信息变得非常接近0或非常接近1。极化本身是RM码在对称信道上实现容量的必要条件，而极化与条件互信息的强有序性一起。信息给出了充分条件，其中强顺序意味着具有较大权重的行总是对应于较大的条件互信息。虽然我们无法证明强序，但我们在条件互信息上建立了一个偏序，它是强序的子集。 虽然本文的主要结果（极化和偏序）为 RM 码的容量实现猜想提供了一些进展，但我们强调我们的结果不允许我们证明该猜想。