Polar Codes’ Simplicity, Random Codes’ Durability

Over any discrete memoryless channel, we build codes such that: for one, their block error probabilities and code rates scale like random codes'; two, encoding decoding complexities polar codes'. Quantitatively, constants $\pi,\rho>0$ that $\pi+2\rho<1$, construct a sequence of correction with length $N$ approaching infinity, probability $\exp(-N^\pi)$, rate $N^{-\rho}$ less than the Shannon capacity, complexity $O(N\log N)$ per block. The putative take uniform $\varsigma$-ary messages sender's choice prime $\varsigma$. are optimal in following manner: Should $\pi+2\rho>1$, no exist generic channels regardless alphabet complexity.